{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(a). 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Calculate the volume of the solid obtained by rotating the region "},{"type":"inlineMath","attrs":{"latex":"0 \\leq y \\leq 1 - \\frac{x^2}{4}"}},{"type":"text","marks":[{"type":"bold"}],"text":" from "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" to "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","marks":[{"type":"bold"}],"text":" about the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Volumes of solids of revolution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Shell method: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{a}^{b} x \\cdot f(x) \\, dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the bounds for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":": $x$ goes from $0"},{"type":"inlineMath","attrs":{"latex":" to $2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", the height of the shell is "},{"type":"inlineMath","attrs":{"latex":"1 - \\frac{x^2}{4}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the shell method integral: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{2} x \\cdot (1 - \\frac{x^2}{4}) \\, dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand and write the integral."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{2} \\left(x - \\frac{x^3}{4}\\right) dx"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluating this integral gives the exact volume. The shell method is appropriate since the region is described in terms of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(c). Calculate the volume of the solid obtained by rotating the region "},{"type":"inlineMath","attrs":{"latex":"0 \\leq y \\leq \\sin x \\cos x"}},{"type":"text","marks":[{"type":"bold"}],"text":" from "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" to "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{4}"}},{"type":"text","marks":[{"type":"bold"}],"text":" about the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Volumes of solids of revolution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the disk method to find the volume when rotating about the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Disk method: "},{"type":"inlineMath","attrs":{"latex":"V = \\pi \\int_{a}^{b} [f(x)]^2 dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the bounds for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":": $x$ goes from $0"},{"type":"inlineMath","attrs":{"latex":" to "}},{"type":"text","text":"\\frac{\\pi}{4}$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The radius of each disk is "},{"type":"inlineMath","attrs":{"latex":"y = \\sin x \\cos x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the disk method integral: "},{"type":"inlineMath","attrs":{"latex":"V = \\pi \\int_{0}^{\\frac{\\pi}{4}} [\\sin x \\cos x]^2 dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify "},{"type":"inlineMath","attrs":{"latex":"[\\sin x \\cos x]^2"}},{"type":"text","text":" using trigonometric identities."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"V = \\pi \\int_{0}^{\\frac{\\pi}{4}} \\sin^2 x \\cos^2 x \\, dx"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the identity "},{"type":"inlineMath","attrs":{"latex":"\\sin^2 x \\cos^2 x = \\frac{1}{4} \\sin^2 2x"}},{"type":"text","text":", you can further simplify the integral before evaluating."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(d). Calculate the volume of the solid obtained by rotating the region "},{"type":"inlineMath","attrs":{"latex":"0 \\leq x \\leq y^2"}},{"type":"text","marks":[{"type":"bold"}],"text":" from "},{"type":"inlineMath","attrs":{"latex":"y = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" to "},{"type":"inlineMath","attrs":{"latex":"y = \\sqrt{2}"}},{"type":"text","marks":[{"type":"bold"}],"text":" about the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Volumes of solids of revolution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the shell method for regions described in terms of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Shell method: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{a}^{b} y \\cdot f(y) \\, dy"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the bounds for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":": $y$ goes from $0"},{"type":"inlineMath","attrs":{"latex":" to "}},{"type":"text","text":"\\sqrt{2}$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":", the shell extends from "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"x = y^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the shell method integral: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{\\sqrt{2}} y \\cdot y^2 \\, dy"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand and write the integral."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{\\sqrt{2}} y^3 \\, dy"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluating this integral gives the exact volume. 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Calculate the volume of the solid obtained by rotating the region "},{"type":"inlineMath","attrs":{"latex":"0 \\leq y \\leq \\frac{9x}{\\sqrt{x^3 + 9}}"}},{"type":"text","marks":[{"type":"bold"}],"text":" from "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" to "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","marks":[{"type":"bold"}],"text":" about the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Volumes of solids of revolution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the shell method for regions described in terms of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Shell method: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{a}^{b} x \\cdot f(x) \\, dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the bounds for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":": $x$ goes from $0"},{"type":"inlineMath","attrs":{"latex":" to $3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", the height of the shell is "},{"type":"inlineMath","attrs":{"latex":"\\frac{9x}{\\sqrt{x^3 + 9}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the shell method integral: "},{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{3} x \\cdot \\frac{9x}{\\sqrt{x^3 + 9}} dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand and write the integral."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"V = 2\\pi \\int_{0}^{3} \\frac{9x^2}{\\sqrt{x^3 + 9}} dx"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluating this integral gives the exact volume. 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Find the arclength of the curve "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{3}{4}x^{4/3} - \\frac{3}{8}x^{2/3} + 5"}},{"type":"text","marks":[{"type":"bold"}],"text":" from "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","marks":[{"type":"bold"}],"text":" to "},{"type":"inlineMath","attrs":{"latex":"x = 8"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Arc length of a curve"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute the length of a curve using the arc length formula."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arc length formula: "},{"type":"inlineMath","attrs":{"latex":"L = \\int_{a}^{b} \\sqrt{1 + [y'(x)]^2} dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"y'(x)"}},{"type":"text","text":" by differentiating "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"[y'(x)]^2"}},{"type":"text","text":" and add $1$ to it."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the arc length integral: "},{"type":"inlineMath","attrs":{"latex":"L = \\int_{1}^{8} \\sqrt{1 + [y'(x)]^2} dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand as much as possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"L = \\int_{1}^{8} \\sqrt{1 + \\left( \\frac{3}{4} \\cdot \\frac{4}{3} x^{1/3} - \\frac{3}{8} \\cdot \\frac{2}{3} x^{-1/3} \\right)^2} dx"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying, the integrand becomes "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{1 + (x^{1/3} - x^{-1/3})^2}"}},{"type":"text","text":". 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Find the surface area of the surface obtained by revolving "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{x^3}{9}"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"0 \\leq x \\leq 2"}},{"type":"text","marks":[{"type":"bold"}],"text":" around the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Surface area of revolution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute the surface area of a solid of revolution using the surface area formula."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Surface area formula: "},{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{a}^{b} f(x) \\sqrt{1 + [f'(x)]^2} dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":" by differentiating "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"[f'(x)]^2"}},{"type":"text","text":" and add $1$ to it."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the surface area integral: "},{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{0}^{2} f(x) \\sqrt{1 + [f'(x)]^2} dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand as much as possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{0}^{2} \\frac{x^3}{9} \\sqrt{1 + \\left( \\frac{1}{3}x^2 \\right)^2} dx"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying, the integrand becomes "},{"type":"inlineMath","attrs":{"latex":"\\frac{x^3}{9} \\sqrt{1 + \\frac{x^4}{9}}"}},{"type":"text","text":". 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Find the surface area of the surface obtained by revolving "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{y^4}{8} + \\frac{1}{8}y^2"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"1 \\leq y \\leq 2"}},{"type":"text","marks":[{"type":"bold"}],"text":" around the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Surface area of revolution (parametric form)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute the surface area of a solid of revolution using the surface area formula for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" as a function of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Surface area formula: "},{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{a}^{b} y \\sqrt{1 + \\left( \\frac{dx}{dy} \\right)^2} dy"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\frac{dx}{dy}"}},{"type":"text","text":" by differentiating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"\\left( \\frac{dx}{dy} \\right)^2"}},{"type":"text","text":" and add $1$ to it."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the surface area integral: "},{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{1}^{2} y \\sqrt{1 + \\left( \\frac{dx}{dy} \\right)^2} dy"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand as much as possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the integral setup before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"S = 2\\pi \\int_{1}^{2} y \\sqrt{1 + \\left( \\frac{1}{2}y^3 + \\frac{1}{4}y \\right)^2} dy"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying, the integrand becomes "},{"type":"inlineMath","attrs":{"latex":"y \\sqrt{1 + (\\frac{1}{2}y^3 + \\frac{1}{4}y)^2}"}},{"type":"text","text":". Evaluate this integral for the surface area."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(a). A force of 2N will stretch a rubber band 2cm. Assuming Hooke’s law applies, how far will a 4N force stretch the rubber band, and what is the work required to do so?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Work and Hooke's Law (Calculus II)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply Hooke's Law and calculate work done in stretching a spring."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Hooke's Law: "},{"type":"inlineMath","attrs":{"latex":"F = kx"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Work: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{a}^{b} F(x) dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the given force and stretch to find the spring constant "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation "},{"type":"inlineMath","attrs":{"latex":"F = kx"}},{"type":"text","text":" for the new force to find the new stretch."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the work integral: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{x} kx dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integral and prepare to evaluate."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the values before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Spring constant "},{"type":"inlineMath","attrs":{"latex":"k = 1"}},{"type":"text","text":" N/cm. A 4N force stretches the band 4cm. The work required is "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{4} x dx = 8"}},{"type":"text","text":" N·cm."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We used Hooke's Law to find "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":", then calculated the work using the integral."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(b). A bag of sand originally weighing 144 lb is lifted at a constant rate. As it rises, sand leaks out at a constant rate. If the sand is half-gone by the time the bag is lifted 18 ft, how much work has been done by lifting the sand this far?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Work with variable force (Calculus II)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and compute work when the force varies as a function of height."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Work: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{a}^{b} F(y) dy"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the weight of the sand as a function of height "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the work integral: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{18} F(y) dy"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the expression for "},{"type":"inlineMath","attrs":{"latex":"F(y)"}},{"type":"text","text":", knowing the sand decreases linearly from 144 lb to 72 lb."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integral and prepare to evaluate."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the values before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"F(y) = 144 - 4y"}},{"type":"text","text":"; "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{18} (144 - 4y) dy = 144 \\times 18 - 2 \\times 18^2 = 2592 - 648 = 1944"}},{"type":"text","text":" ft·lb."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The force decreases linearly, so the work is the area under the force curve."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(c). An electric elevator with a motor at the top has a multistrand cable weighing 4.5 lb/ft. When the car is at the first floor, 180 ft of cable are paid out. When the car is at the top, zero feet are out. How much work does the motor do on the cable when it takes the car from the first floor to the top?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Work with variable force (Calculus II)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and compute work when the force varies as a function of height."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Work: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{a}^{b} F(y) dy"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the weight of the cable as a function of height "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the work integral: "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{180} F(y) dy"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the expression for "},{"type":"inlineMath","attrs":{"latex":"F(y)"}},{"type":"text","text":", knowing the length of cable decreases as the elevator rises."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integral and prepare to evaluate."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to compute the values before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"F(y) = 4.5 \\times (180 - y)"}},{"type":"text","text":"; "},{"type":"inlineMath","attrs":{"latex":"W = \\int_{0}^{180} 4.5(180 - y) dy = 4.5 \\times 180 \\times 180 - 4.5 \\times \\frac{180^2}{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluating gives "},{"type":"inlineMath","attrs":{"latex":"W = 145800 - 72900 = 72900"}},{"type":"text","text":" ft·lb."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(a). Rewrite as a single logarithm: "},{"type":"inlineMath","attrs":{"latex":"\\ln \\sin \\theta - \\ln \\left( \\sin \\theta / 5 \\right)"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithm properties"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to combine logarithmic expressions using properties of logarithms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\ln a - \\ln b = \\ln \\left( \\frac{a}{b} \\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the subtraction property of logarithms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the argument inside the logarithm."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to combine the logs before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\ln \\sin \\theta - \\ln \\left( \\frac{\\sin \\theta}{5} \\right) = \\ln 5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The logs combine to a single logarithm of 5."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(b). Rewrite as a single logarithm: "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{2} \\ln(4t^4) - \\ln 2"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithm properties"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to combine logarithmic expressions using properties of logarithms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"a \\ln b = \\ln b^a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\ln a - \\ln b = \\ln \\left( \\frac{a}{b} \\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{2} \\ln(4t^4)"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\ln(4t^4)^{1/2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the argument inside the logarithm."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the logs using subtraction property."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to combine the logs before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{1}{2} \\ln(4t^4) - \\ln 2 = \\ln \\left( \\frac{2t^2}{2} \\right) = \\ln t^2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The logs combine to a single logarithm of "},{"type":"inlineMath","attrs":{"latex":"t^2"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(c). Rewrite as a single logarithm: "},{"type":"inlineMath","attrs":{"latex":"3 \\ln \\sqrt{t^2 - 1} - \\ln(t + 1)"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithm properties"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to combine logarithmic expressions using properties of logarithms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"a \\ln b = \\ln b^a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\ln a - \\ln b = \\ln \\left( \\frac{a}{b} \\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"3 \\ln \\sqrt{t^2 - 1}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\ln (\\sqrt{t^2 - 1})^3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the argument inside the logarithm."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the logs using subtraction property."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pause here: Try to combine the logs before proceeding to the final calculation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"3 \\ln \\sqrt{t^2 - 1} - \\ln(t + 1) = \\ln \\left( \\frac{(t^2 - 1)^{3/2}}{t + 1} \\right)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The logs combine to a single logarithm of "},{"type":"inlineMath","attrs":{"latex":"\\frac{(t^2 - 1)^{3/2}}{t + 1}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(a). Calculate the derivative: "},{"type":"inlineMath","attrs":{"latex":"y = (x^2 \\ln x)^4"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Differentiation (chain rule, product rule)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to differentiate composite and product functions."}]},{"type":"heading","attrs":{"textAli