Comprehensive Calculus II Exam Review: Derivatives, Curve Sketching, and Analysis

Q1. Find the derivatives of the following functions. You do not need to simplify your final answers.

Background

Topic: Differentiation Rules

This question tests your ability to apply the basic rules of differentiation, including the power rule, product rule, quotient rule, chain rule, and derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions.

Key Terms and Formulas:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

• Derivatives of basic functions (e.g., $\sin x$, $\cos x$, $e^x$, $\ln x$, $\arcsin x$, etc.)

Step-by-Step Guidance

1. For each function, first identify which rule(s) apply: Is it a product, quotient, or composition of functions?

2. Write down the function and label $u(x)$ and $v(x)$ if using product or quotient rule.

3. Apply the appropriate differentiation rule(s) step by step, showing each derivative taken.

4. For composite functions, use the chain rule and clearly indicate the inner and outer functions.

5. For higher-order derivatives, repeat the differentiation process as needed.

Try solving on your own before revealing the answer!

Q2. Find all x-values at which $f(x) = \frac{\sqrt[3]{x}-1}{12x}$ has a horizontal tangent line.

Background

Topic: Critical Points and Tangent Lines

This question is about finding where the derivative of a function equals zero, which corresponds to horizontal tangent lines on the graph of the function.

Key Terms and Formulas:

• Horizontal tangent line: Occurs where $f'(x) = 0$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

Step-by-Step Guidance

1. Let $u(x) = \sqrt[3]{x} - 1$ and $v(x) = 12x$.

2. Find $u'(x)$ and $v'(x)$ separately.

3. Apply the quotient rule to find $f'(x)$.

4. Set $f'(x) = 0$ and solve for $x$ to find where the tangent is horizontal.

Try solving on your own before revealing the answer!

Q3. Find the second derivative of $f(x) = \sec x$.

Background

Topic: Higher-Order Derivatives

This question asks you to find the second derivative, which means differentiating the function twice. This is useful for analyzing concavity and points of inflection.

Key Terms and Formulas:

• First derivative of $\sec x$: $\frac{d}{dx} \sec x = \sec x \tan x$

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Derivative of $\tan x$: $\frac{d}{dx} \tan x = \sec^2 x$

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = \sec x \tan x$.

2. Apply the product rule to $f'(x)$ to find $f''(x)$.

3. Differentiate $\sec x$ and $\tan x$ as needed in the product rule.

4. Combine like terms and simplify as much as possible, but do not fully expand or combine all terms yet.

Try solving on your own before revealing the answer!

Q4. Find the fourth derivative of $f(x) = x^4 + e^{2x} + \ln(x^5)$.

Background

Topic: Higher-Order Derivatives

This question tests your ability to repeatedly differentiate a function, using the rules for polynomials, exponentials, and logarithms.

Key Terms and Formulas:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Derivative of $e^{ax}$: $\frac{d}{dx} e^{ax} = a e^{ax}$

• Derivative of $\ln(x^n)$: $\frac{d}{dx} \ln(x^n) = \frac{n}{x}$

Step-by-Step Guidance

1. Find the first derivative of each term separately.

2. Continue differentiating each term until you reach the fourth derivative.

3. For the polynomial, note that after a certain number of derivatives, the term will become zero.

4. For the exponential and logarithmic terms, apply the rules for each differentiation step.

Try solving on your own before revealing the answer!

Q5. Find at least three different ways (by algebraically rewriting the function) to evaluate the derivative of $y = \frac{(3x+5)^2}{x^2}$, and simplify each derivative to show that they are all equivalent.

Background

Topic: Algebraic Manipulation and Differentiation

This question is about recognizing that a function can be rewritten in different forms, and that the derivative should be the same regardless of the form used. It reinforces understanding of the product, quotient, and chain rules.

Key Terms and Formulas:

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Rewrite $y$ in at least three different algebraic forms (e.g., expand numerator, write as a product, or use negative exponents).

2. For each form, apply the appropriate differentiation rule (product, quotient, or chain rule).

3. Simplify each derivative as much as possible to show equivalence.

4. Compare the simplified derivatives to confirm they are the same.

Try solving on your own before revealing the answer!

Q6. Use technology to graph the functions $f(x) = \arccos x$ and its first two derivatives. Compare and contrast the three graphs - what do they all have in common? What features of $y = f(x)$ are related to features of $y = f'(x)$, and how? What about $y = f(x)$ vs $y = f''(x)$? Or $y = f'(x)$ vs $y = f''(x)$?

Background

Topic: Graphical Analysis of Functions and Their Derivatives

This question asks you to interpret and compare the graphs of a function and its derivatives, focusing on how features such as maxima, minima, and inflection points relate between the graphs.

Key Terms and Concepts:

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

• Inflection points: Where $f''(x) = 0$ or changes sign

• Relationship between $f(x)$, $f'(x)$, and $f''(x)$

Step-by-Step Guidance

1. Graph $f(x) = \arccos x$ and its first two derivatives using graphing technology.

2. Identify key features on each graph: maxima, minima, points of inflection, and intervals of increase/decrease.

3. Compare where $f(x)$ has horizontal tangents to where $f'(x)$ crosses the x-axis.

4. Discuss how the concavity of $f(x)$ relates to the sign of $f''(x)$.

Try analyzing the graphs and relationships before reading the sample explanations!

Q7. Sketch a function with the following characteristics: Increasing on $(-\infty,-2)$, $(4,\infty)$; decreasing on $(-2,4)$; concave up on $(1,\infty)$; concave down on $(-\infty,1)$; local maximum at $(-2,8)$; local minimum at $(4,-10)$, point of inflection at $(1,-1)$.

Background

Topic: Curve Sketching

This question tests your ability to synthesize information about increasing/decreasing intervals, concavity, extrema, and inflection points to sketch a function.

Key Terms and Concepts:

• Increasing/Decreasing Intervals: Determined by the sign of $f'(x)$

• Concavity: Determined by the sign of $f''(x)$

• Local Maximum/Minimum: Where $f'(x) = 0$ and $f''(x)$ indicates concavity

• Point of Inflection: Where $f''(x)$ changes sign

Step-by-Step Guidance

1. Mark the given points: local max at $(-2,8)$, local min at $(4,-10)$, inflection at $(1,-1)$.

2. Draw the function increasing on $(-\infty,-2)$ and $(4,\infty)$, decreasing on $(-2,4)$.

3. Ensure the graph is concave down on $(-\infty,1)$ and concave up on $(1,\infty)$, with a change at $(1,-1)$.

4. Connect the points smoothly, respecting the required behavior.

Try sketching the graph before checking a sample!

Q8. For each of the following functions, state any intervals on which the graph is increasing, decreasing, concave up, and concave down. List all local extrema and points of inflection. Use this information, along with any additional information about the domain, intercepts, or asymptotes of the function to draw a graph.

Background

Topic: Curve Sketching and Analysis

This question asks you to analyze a function using calculus to determine its behavior and sketch its graph. You will use the first and second derivatives to find intervals of increase/decrease, concavity, extrema, and inflection points.

Key Terms and Formulas:

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

• Intervals of increase/decrease: Determined by the sign of $f'(x)$

• Concavity: Determined by the sign of $f''(x)$

• Points of inflection: Where $f''(x)$ changes sign

• Domain, intercepts, asymptotes

Step-by-Step Guidance

1. Find the first derivative and solve $f'(x) = 0$ to locate critical points.

2. Test intervals between critical points to determine where the function is increasing or decreasing.

3. Find the second derivative and solve $f''(x) = 0$ to locate possible inflection points.

4. Test intervals between inflection points to determine concavity.

5. Use all this information to sketch the graph, marking intercepts, asymptotes, and key points.

Try analyzing and sketching each function before checking the sample graphs!

Q9. Use a limit and l’Hôpital’s Rule for the following:

• (a) Describe the behavior of $y = \frac{\cos x - 1}{x}$ around $x = 0$.

• (b) Describe the end behavior of $y = \frac{3x^4}{5x^2 - 2x + 7}$ as $x \to \infty$.

• (c) Describe the end behavior of $y = t e^{-t}$ as $t \to \infty$.

• (d) Describe the behavior of $y = \sin x \ln x$ as $x \to 0^+$.

• (e) Describe the behavior of $y = (1-2x)^{1/x}$ as $x \to 0^+$.

Background

Topic: Limits and L’Hôpital’s Rule

This question tests your ability to evaluate limits that result in indeterminate forms using L’Hôpital’s Rule, which involves differentiating the numerator and denominator until the limit can be evaluated.

Key Terms and Formulas:

• L’Hôpital’s Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit exists).

• Indeterminate forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $1^\infty$, etc.

Step-by-Step Guidance

1. For each limit, first substitute the value to check if it is indeterminate.

2. If indeterminate, apply L’Hôpital’s Rule by differentiating numerator and denominator as needed.

3. Repeat the process if the result is still indeterminate.

4. Interpret the result in terms of the behavior of the function near the point or as $x \to \infty$.

Try working through the limits before checking the solutions!