Q1. Select the graph corresponding to

Background

Topic: Exponential Functions and Graphs

This question tests your understanding of how exponential functions are transformed and how to interpret their graphs.

Key Terms and Formulas:

• Exponential Decay: decreases as increases.

• Vertical Shift: The function is shifted up by 5 units and reflected over the x-axis compared to .

Step-by-Step Guidance

1. Recall the basic shape of : it starts high on the left and decreases toward zero as increases.

2. Consider the effect of the negative sign: reflects the graph over the x-axis.

3. Add 5 to the function: shifts the entire graph up by 5 units.

4. Think about the y-intercept: plug in to find at that point.

5. Analyze the end behavior as and to match the correct graph.

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Q2. Find if

Background

Topic: Differentiation (Product, Chain, and Logarithmic Differentiation)

This question tests your ability to find the second derivative of a function involving both a variable base and exponent.

Key Terms and Formulas:

• Logarithmic Differentiation: Useful for functions of the form .

• Chain Rule:

• Product Rule:

Step-by-Step Guidance

1. Let . Take the natural logarithm of both sides to simplify differentiation.

2. Use implicit differentiation to find , applying the chain rule and product rule as needed.

3. After finding , express it in terms of (and possibly ).

4. To find , differentiate again with respect to , using the product and chain rules as necessary.

5. Set up the expression for , but do not simplify or compute the final value yet.

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Q3. Find the intervals in which the function is concave up/concave down and its inflection points.

Background

Topic: Concavity and Inflection Points

This question tests your ability to determine where a function is concave up or down by analyzing the sign of its second derivative, and to find inflection points.

Key Terms and Formulas:

• Concavity: A function is concave up where and concave down where .

• Inflection Point: A point where changes sign.

• Second Derivative:

Step-by-Step Guidance

1. Find the first derivative using the chain rule.

2. Find the second derivative by differentiating .

3. Set and solve for to find possible inflection points.

4. Test intervals around these points to determine where is positive (concave up) or negative (concave down).

5. List the intervals of concavity and identify the inflection points based on sign changes.

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Q4. Find the equation of the tangent line at the point (1, 2) on the curve .

Background

Topic: Implicit Differentiation and Tangent Lines

This question tests your ability to use implicit differentiation to find the slope of a curve at a given point and write the equation of the tangent line.

Key Terms and Formulas:

• Implicit Differentiation: Used when is defined implicitly in terms of .

• Tangent Line Equation: , where is the slope at .

Step-by-Step Guidance

1. Differentiate both sides of the equation with respect to , treating as a function of $x$ (use the chain rule for $y$ terms).

2. Solve for to find the slope of the tangent line at any point .

3. Plug in and to find the specific slope at the given point.

4. Use the point-slope form to write the equation of the tangent line, but do not substitute the final slope value yet.

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Q5. A part of a roof is an isosceles triangle ABC with AC = BC. A rectangular region PQRS has to be selected to install solar panels. Given AB = 24 ft and CE = 10 ft, let PQ = 2x and PS = y. What values of x and y optimize the rectangle PQRS?

Background

Topic: Optimization (Constrained Maximization)

This question tests your ability to set up and solve an optimization problem with geometric constraints.

Key Terms and Formulas:

• Area of Rectangle:

• Constraint: The rectangle must fit inside the triangle, so and are related by the triangle's dimensions.

Step-by-Step Guidance

1. Express the area of the rectangle in terms of and .

2. Use the geometry of the triangle to find a relationship (constraint) between and .

3. Substitute the constraint into the area formula to write as a function of a single variable.

4. Take the derivative of with respect to that variable and set it to zero to find critical points.

5. Set up the equation to solve for the optimal values of and , but do not solve for the final values yet.

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Q6. An axis-parallel rectangle is formed with one corner at (0, 0) and its opposite corner at the point (a, b) on the line . Find the dimensions of the rectangle whose area is maximized when (a, b) lies on the line.

Background

Topic: Optimization (Area Maximization with Linear Constraint)

This question tests your ability to maximize the area of a rectangle under a linear constraint.

Key Terms and Formulas:

• Area of Rectangle:

• Constraint:

Step-by-Step Guidance

1. Express the area in terms of and .

2. Substitute the constraint into the area formula to get as a function of only.

3. Take the derivative of with respect to and set it to zero to find the critical point.

4. Solve for and then use the constraint to find .

5. Set up the equations for the optimal dimensions, but do not compute the final values yet.

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Q7. A 5-foot-tall actress is 12 feet from a pole and moving away at 2 ft/s. At that moment, she casts a 6-foot shadow, and the shadow is increasing at 2 ft/s. Determine the height of the spotlight and the rate at which the height must be changing at this moment.

Background

Topic: Related Rates

This question tests your ability to set up and solve a related rates problem involving similar triangles and rates of change.

Key Terms and Formulas:

• Similar Triangles: Used to relate the height of the pole, the length of the shadow, and the distance from the pole.

• Related Rates: Differentiate both sides of the equation with respect to time .

Step-by-Step Guidance

1. Draw a diagram and label all known quantities: height of actress, distance from pole, shadow length, and rates of change.

2. Set up a proportion using similar triangles to relate the height of the spotlight to the other quantities.

3. Differentiate both sides of the equation with respect to time to relate the rates of change.

4. Plug in the known values for distances and rates at the given moment.

5. Set up the equation to solve for the height of the spotlight and its rate of change, but do not compute the final values yet.

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Q8. Air is pushed from a cylindrical pump (diameter 4 in, height 12 in, decreasing at 4 in/min) into a spherical balloon (diameter 6 in). Find the rate at which the diameter of the sphere is changing at that moment.

Background

Topic: Related Rates (Volumes of Cylinder and Sphere)

This question tests your ability to relate the rates of change of volumes and dimensions for different shapes.

Key Terms and Formulas:

• Volume of Cylinder:

• Volume of Sphere:

• Related Rates: (air leaving pump enters balloon)

Step-by-Step Guidance

1. Find the rate at which the volume of air in the pump is decreasing using the given dimensions and rate.

2. Set this equal to the rate at which the volume of the balloon is increasing.

3. Express in terms of , where is the radius of the balloon.

4. Plug in the known values for the radius and solve for .

5. Relate to the rate of change of the diameter, but do not compute the final value yet.

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Q9. Find .

Background

Topic: Integration (Definite Integrals)

This question tests your ability to compute definite integrals involving polynomials and exponentials.

Key Terms and Formulas:

• Integral of :

• Integral of :

• Definite Integral: , where is an antiderivative of .

Step-by-Step Guidance

1. Find the antiderivative of .

2. Evaluate the antiderivative at the upper limit and the lower limit .

3. Subtract the value at from the value at to get the definite integral.

4. Set up the expression for the final answer, but do not compute the numeric value yet.

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Q10. (a) Express with sigma notation the left Riemann sum of on using 167 rectangles. (b) Determine geometrically.

Background

Topic: Riemann Sums and Geometric Interpretation of Integrals

This question tests your understanding of Riemann sums and the geometric meaning of definite integrals.

Key Terms and Formulas:

• Left Riemann Sum:

• Width of Each Rectangle:

• Geometric Area: The area under from to is a triangle.

Step-by-Step Guidance

1. For (a): Find and write the left endpoints in terms of .

2. Write the left Riemann sum in sigma notation using and .

3. For (b): Sketch the region under from to and recognize it as a triangle.

4. Set up the formula for the area of the triangle, but do not compute the final value yet.

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Q11. Given marginal cost and marginal revenue , determine the total change in profit between and .

Background

Topic: Marginal Analysis and Integrals

This question tests your ability to use marginal cost and revenue functions to find the change in profit over an interval.

Key Terms and Formulas:

• Marginal Cost: is the derivative of the cost function.

• Marginal Revenue: is the derivative of the revenue function.

• Change in Profit:

Step-by-Step Guidance

1. Set up the integral for the change in profit: .

2. Substitute the given functions for and into the integral.

3. Simplify the integrand as much as possible.

4. Set up the expression for the definite integral, but do not compute the final value yet.

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Q12. Find the area of the region bounded by and on the interval . No need to simplify your final answer.

Background

Topic: Area Between Curves

This question tests your ability to set up and evaluate the definite integral representing the area between two curves.

Key Terms and Formulas:

• Area Between Curves: , where on .

Step-by-Step Guidance

1. Determine which function is on top (greater) over the interval .

2. Set up the integral for the area: .

3. Write the explicit expression for the integrand using the given functions.

4. Set up the definite integral, but do not evaluate or simplify it yet.

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