Step-by-Step Calculus Study Guidance for Practice Final Exam

Q2. Find the equation of the tangent line to $p(x) = 4x^2 + 2x^4 + \frac{4}{x^3} + 4$ at $x = 1$.

Background

Topic: Tangent Lines & Derivatives

This question tests your ability to find the equation of a tangent line to a function at a specific point using derivatives.

Key Terms and Formulas:

• Tangent line: A line that touches a curve at a single point and has the same slope as the curve at that point.

• Derivative: $p'(x)$ gives the slope of the tangent line at $x$.

• Point-slope form: $y = p'(a)(x - a) + p(a)$ for tangent at $x = a$.

Step-by-Step Guidance

1. Find $p'(x)$ by differentiating $p(x)$ term by term.

2. Evaluate $p'(1)$ to get the slope of the tangent line at $x = 1$.

3. Calculate $p(1)$ to find the y-coordinate of the point of tangency.

4. Set up the tangent line equation using the point-slope form: $y = p'(1)(x - 1) + p(1)$.

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Q6. Find the derivative of $q(x) = \frac{3x^3 + 9}{5x^2 + 6}$.

Background

Topic: Derivatives & Quotient Rule

This question tests your ability to use the quotient rule to differentiate a rational function.

Key Terms and Formulas:

• Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Here, $u = 3x^3 + 9$, $v = 5x^2 + 6$

Step-by-Step Guidance

1. Identify $u$ and $v$ and compute $u'$ and $v'$.

2. Apply the quotient rule formula to $q(x)$.

3. Expand and simplify the numerator as much as possible.

4. Write the derivative in the form $\frac{\text{numerator}}{(5x^2 + 6)^2}$.

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Q7. (a) Find the time when an object thrown straight up reaches its highest point, given $h(t) = -32t^2 + 75t^2 + 173$.

Background

Topic: Optimization & Maximum/Minimum Problems

This question tests your ability to find the maximum value of a quadratic function, which models the height of an object.

Key Terms and Formulas:

• Maximum of a parabola: Occurs at $t = -\frac{b}{2a}$ for $h(t) = at^2 + bt + c$.

• Critical point: Where $h'(t) = 0$.

Step-by-Step Guidance

1. Find $h'(t)$ by differentiating $h(t)$ with respect to $t$.

2. Set $h'(t) = 0$ and solve for $t$ to find the time of maximum height.

3. Check that the second derivative $h''(t)$ is negative to confirm a maximum.

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Q10. Find the absolute maximum value of $f(x) = 2x^3 - 30x^2 + 126x + 9$ on $[4,8]$.

Background

Topic: Absolute Extrema & Optimization

This question tests your ability to find the absolute maximum of a function on a closed interval using calculus.

Key Terms and Formulas:

• Critical points: Where $f'(x) = 0$ or $f'(x)$ is undefined.

• Absolute maximum: Largest value among $f(a)$, $f(b)$, and $f$ at critical points in $[a, b]$.

Step-by-Step Guidance

1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points in $[4,8]$.

2. Evaluate $f(x)$ at the endpoints $x = 4$ and $x = 8$.

3. Evaluate $f(x)$ at any critical points found in the interval.

4. Compare all values to determine which is the absolute maximum.

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Q13. Find the critical point of $f(x) = 5e^{0.2x^2 - 3x}$.

Background

Topic: Critical Points & Derivatives

This question tests your ability to find where the derivative of a function is zero (critical points).

Key Terms and Formulas:

• Critical point: Where $f'(x) = 0$.

• Chain rule: Used to differentiate composite functions.

Step-by-Step Guidance

1. Find $f'(x)$ using the chain rule.

2. Set $f'(x) = 0$ and solve for $x$.

3. Check if the solution is a valid critical point.

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Q17. Evaluate $\lim_{x \to -2} \frac{\sqrt{x + 27} - 5}{x + 2}$.

Background

Topic: Limits & Conjugate Method

This question tests your ability to evaluate a limit involving a square root by rationalizing the numerator.

Key Terms and Formulas:

• Conjugate: Multiply numerator and denominator by $\sqrt{x + 27} + 5$.

• Limit: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$.

Step-by-Step Guidance

1. Multiply numerator and denominator by the conjugate $\sqrt{x + 27} + 5$.

2. Simplify the numerator using $(a - b)(a + b) = a^2 - b^2$.

3. Simplify the denominator and substitute $x = -2$.

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Q23. Find $\frac{dy}{dx}$ by implicit differentiation for $4x^2 + 8xy + 2y^2 = 15$.

Background

Topic: Implicit Differentiation

This question tests your ability to differentiate equations where $y$ is a function of $x$ but not isolated.

Key Terms and Formulas:

• Implicit differentiation: Differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Chain rule: $\frac{d}{dx} y^2 = 2y \frac{dy}{dx}$

Step-by-Step Guidance

1. Differentiate each term in the equation with respect to $x$.

2. Collect all terms involving $\frac{dy}{dx}$ on one side.

3. Solve for $\frac{dy}{dx}$ in terms of $x$ and $y$.

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Q25. Find the domain of $f(x, y) = \sqrt{x} + \sqrt{y}$.

Background

Topic: Domain of Multivariable Functions

This question tests your understanding of the domain for functions involving square roots.

Key Terms and Formulas:

• Domain: Set of all $(x, y)$ where the function is defined.

• Square root: Defined only for non-negative arguments.

Step-by-Step Guidance

1. Set $x \geq 0$ and $y \geq 0$ for the square roots to be defined.

2. Express the domain as the set of all $(x, y)$ in the $xy$-plane where $x \geq 0$ and $y \geq 0$.

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Q28. Find the horizontal asymptote of $f(x) = \frac{2x - 4}{4x^2 + 5x - 2}$.

Background

Topic: Asymptotes of Rational Functions

This question tests your ability to find horizontal asymptotes by comparing degrees of numerator and denominator.

Key Terms and Formulas:

• Horizontal asymptote: $y = 0$ if degree of numerator < denominator.

• If degrees are equal, asymptote is $y = \frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}}$.

Step-by-Step Guidance

1. Compare degrees of numerator and denominator.

2. Determine the horizontal asymptote based on degree comparison.

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Q31. If $f(-4) = f(4)$ for a differentiable function $f(x)$, what does the Mean Value Theorem guarantee?

Background

Topic: Mean Value Theorem

This question tests your understanding of the Mean Value Theorem for differentiable functions.

Key Terms and Formulas:

• Mean Value Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then $\exists c \in (a, b)$ with $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Step-by-Step Guidance

1. Apply the Mean Value Theorem to $f(x)$ on $[-4, 4]$.

2. Since $f(-4) = f(4)$, calculate $f'(c)$ using the theorem.

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Q32. Find $f^{(3)}(x)$ for $f(x) = 3 + 2x^2 + e^{5x}$.

Background

Topic: Higher Order Derivatives

This question tests your ability to compute the third derivative of a function.

Key Terms and Formulas:

• Derivative rules: Power rule, exponential rule.

• Third derivative: $f^{(3)}(x)$ is the derivative of $f''(x)$.

Step-by-Step Guidance

1. Find $f'(x)$ by differentiating each term.

2. Find $f''(x)$ by differentiating $f'(x)$.

3. Find $f^{(3)}(x)$ by differentiating $f''(x)$.

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Q33. Find the interval where $f(t) = -2t^3 + 27t^2 - 108t + 120$ is increasing.

Background

Topic: Increasing/Decreasing Intervals

This question tests your ability to find where a function is increasing by analyzing its derivative.

Key Terms and Formulas:

• Increasing interval: Where $f'(t) > 0$.

• Find critical points by solving $f'(t) = 0$.

Step-by-Step Guidance

1. Find $f'(t)$ and solve $f'(t) = 0$ for critical points.

2. Test intervals between critical points to see where $f'(t) > 0$.

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Q34. Evaluate $\lim_{x \to 7} \frac{2x^3 - 18x^2 + 28x}{x^2 - 11x + 28}$ rounded to 4 decimal places.

Background

Topic: Limits & Factoring

This question tests your ability to evaluate a limit by factoring and simplifying the expression.

Key Terms and Formulas:

• Factoring: Simplify numerator and denominator if possible.

• Direct substitution: Substitute $x = 7$ after simplification.

Step-by-Step Guidance

1. Factor numerator and denominator if possible.

2. Simplify the expression and substitute $x = 7$.

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Q35. Evaluate $\lim_{x \to \infty} \frac{2x^3 + 6x^2 + 3x - 6}{2x^2 - 4x + 15}$.

Background

Topic: Limits at Infinity

This question tests your ability to evaluate limits as $x$ approaches infinity for rational functions.

Key Terms and Formulas:

• Limits at infinity: Compare degrees of numerator and denominator.

• If degree numerator > denominator, limit is infinity or does not exist.

• If degrees are equal, limit is ratio of leading coefficients.

Step-by-Step Guidance

1. Identify degrees of numerator and denominator.

2. Divide numerator and denominator by highest power of $x$ in denominator.

3. Take the limit as $x \to \infty$.

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Q36. Using the definition of the derivative, find $f'(x)$ for $f(x) = 3(x + 2)^2$.

Background

Topic: Definition of the Derivative

This question tests your ability to use the limit definition to find the derivative.

Key Terms and Formulas:

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Step-by-Step Guidance

1. Write $f(x + h)$ and $f(x)$.

2. Set up the difference quotient $\frac{f(x + h) - f(x)}{h}$.

3. Simplify the numerator and take the limit as $h \to 0$.

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Q41. Find the derivative of $h(x) = \frac{e^{-3}}{3 + e^{(2x^2 + 10)}}$.

Background

Topic: Derivatives & Quotient Rule (Exponential Functions)

This question tests your ability to use the quotient rule and chain rule for exponential functions.

Key Terms and Formulas:

• Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Chain rule: Differentiate $e^{(2x^2 + 10)}$.

Step-by-Step Guidance

1. Identify $u = e^{-3}$ and $v = 3 + e^{(2x^2 + 10)}$.

2. Compute $u'$ and $v'$ using chain rule.

3. Apply the quotient rule formula.

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Q42. Find the domain of $g(x) = \frac{88 \sqrt{x - 5}}{15x^2 - 75x + 90}$.

Background

Topic: Domain of Rational Functions with Square Roots

This question tests your ability to find the domain of a function with a square root in the numerator and a quadratic in the denominator.

Key Terms and Formulas:

• Domain: Values of $x$ where $g(x)$ is defined.

• Square root: $x - 5 \geq 0$.

• Denominator: $15x^2 - 75x + 90 \neq 0$.

Step-by-Step Guidance

1. Set $x - 5 \geq 0$ for the square root.

2. Solve $15x^2 - 75x + 90 = 0$ for values to exclude from the domain.

3. Combine the conditions to write the domain in interval notation.

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