Calculus I Practice Test Study Guidance

Q1. Evaluate the following limit using L'Hôpital's Rule when it is convenient and applicable:

$\lim\limits_{x \to 2} \frac{x^2 - 4}{\sqrt{x+2} - 2}$

Background

Topic: Limits and L'Hôpital's Rule

This question tests your ability to evaluate limits that result in indeterminate forms (like $0/0$) and to apply L'Hôpital's Rule, which involves differentiating the numerator and denominator.

Key Terms and Formulas:

• L'Hôpital's Rule: If $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ results in $0/0$ or $\infty/\infty$, then $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit on the right exists).

• Indeterminate Form: A limit that initially gives $0/0$ or $\infty/\infty$.

Step-by-Step Guidance

1. First, substitute $x = 2$ into the numerator and denominator to check if you get an indeterminate form.

2. If you get $0/0$, confirm that L'Hôpital's Rule applies.

3. Differentiate the numerator $f(x) = x^2 - 4$ and the denominator $g(x) = \sqrt{x+2} - 2$ with respect to $x$.

4. Set up the new limit using the derivatives: $\lim\limits_{x \to 2} \frac{f'(x)}{g'(x)}$.

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Q2. Determine whether Rolle's Theorem applies to the given function on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem:

$f(x) = x^2 - 4x + 4$ on $[0, 4]$

Background

Topic: Rolle's Theorem

This question tests your understanding of the conditions for Rolle's Theorem and your ability to check if a function meets those conditions on a closed interval.

Key Terms and Formulas:

• Rolle's Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f'(c) = 0$.

Step-by-Step Guidance

1. Check if $f(x)$ is continuous on $[0, 4]$ and differentiable on $(0, 4)$.

2. Evaluate $f(0)$ and $f(4)$ to see if $f(0) = f(4)$.

3. If all conditions are met, find $c$ such that $f'(c) = 0$.

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Q3. A rectangular pen is built with one side against a barn. If 48 ft of fencing are used for the other three sides of the pen, what dimensions maximize the area?

Background

Topic: Optimization (Applications of Derivatives)

This question tests your ability to set up and solve an optimization problem using calculus, specifically maximizing area with a constraint on perimeter.

Key Terms and Formulas:

• Area of a rectangle: $A = x \cdot y$

• Perimeter constraint: $x + 2y = 48$ (since one side is against the barn)

Step-by-Step Guidance

1. Let $x$ be the length parallel to the barn and $y$ be the width (perpendicular to the barn).

2. Write the constraint equation: $x + 2y = 48$.

3. Solve for one variable in terms of the other using the constraint.

4. Substitute into the area formula to get $A$ as a function of one variable.

5. Find the critical points by taking the derivative and setting it to zero.

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Q4. Determine the intervals on which the following function is concave up or concave down. Select the correct choice and, if necessary, fill in the answer boxes to complete your choice:

$g(x) = x^4 - 5x^2 + 3$

Background

Topic: Concavity and Inflection Points

This question tests your ability to use the second derivative to determine where a function is concave up or down and to find inflection points.

Key Terms and Formulas:

• Second Derivative Test: $g''(x) > 0$ means concave up; $g''(x) < 0$ means concave down.

• Inflection Point: Where $g''(x)$ changes sign.

Step-by-Step Guidance

1. Find the first derivative $g'(x)$ and then the second derivative $g''(x)$.

2. Solve $g''(x) = 0$ to find possible inflection points.

3. Test intervals between these points to determine where $g''(x)$ is positive or negative.

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Q5. Determine the absolute maximum/minimum of the given interval for $f(x) = x^3 - 3x^2 + 2$ on $[0, 3]$

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the absolute maximum and minimum values of a function on a closed interval using critical points and endpoints.

Key Terms and Formulas:

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

• Absolute Extrema: Evaluate $f(x)$ at critical points and endpoints, then compare values.

Step-by-Step Guidance

1. Find $f'(x)$ and solve $f'(x) = 0$ to get critical points in $[0, 3]$.

2. Evaluate $f(x)$ at each critical point and at the endpoints $x = 0$ and $x = 3$.

3. Compare all values to determine the absolute maximum and minimum.

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Q6. Use the graph to identify the points of the interval $[a, b]$ at which local and absolute extrema values exist.

Background

Topic: Extrema from Graphs

This question tests your ability to read a graph and identify where local and absolute maximum and minimum values occur.

Key Terms and Formulas:

• Local Maximum/Minimum: Highest/lowest point in a small neighborhood.

• Absolute Maximum/Minimum: Highest/lowest point on the entire interval.

Step-by-Step Guidance

1. Examine the graph for peaks (local maxima) and valleys (local minima).

2. Compare all values on the interval to determine the absolute maximum and minimum.

3. List the $x$-values where these extrema occur.

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Q7. Determine whether the Mean Value Theorem applies to the function $f(x) = \sqrt{x}$ on the interval $[0, 4]$. If so, find the value(s) of $c$ that are guaranteed to exist by the Mean Value Theorem.

Background

Topic: Mean Value Theorem (MVT)

This question tests your understanding of the Mean Value Theorem and your ability to check if a function meets the necessary conditions on a closed interval.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Check if $f(x) = \sqrt{x}$ is continuous on $[0, 4]$ and differentiable on $(0, 4)$.

2. If so, set up the equation $f'(c) = \frac{f(4) - f(0)}{4 - 0}$.

3. Solve for $c$ in $(0, 4)$.

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Q8. Determine whether the Mean Value Theorem applies to the function $f(x) = \sqrt{x}$ on the interval $[-1, 4]$. If so, find the value(s) of $c$ that are guaranteed to exist by the Mean Value Theorem.

Background

Topic: Mean Value Theorem (MVT)

This question tests your ability to check the domain of a function and the conditions for the Mean Value Theorem.

Key Terms and Formulas:

• Domain of $\sqrt{x}$: $x \geq 0$

• Mean Value Theorem: Same as above.

Step-by-Step Guidance

1. Check if $f(x) = \sqrt{x}$ is defined and continuous on $[-1, 4]$.

2. If not, explain why the Mean Value Theorem does not apply.

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Q9. Write the formula for Newton's method and use the given initial approximation to compute the approximation, if any, for $x^3 - 2x - 5 = 0$ with $x_1 = 2$.

Background

Topic: Newton's Method for Approximating Roots

This question tests your ability to use Newton's Method to approximate solutions to equations numerically.

Key Terms and Formulas:

• Newton's Method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Step-by-Step Guidance

1. Write $f(x) = x^3 - 2x - 5$ and find $f'(x)$.

2. Plug $x_1 = 2$ into the Newton's Method formula to find $x_2$.

3. Set up the calculation for $x_2$ but do not compute the final value yet.

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Q10. Find the intervals on which $f(x) = x^3/x^2$ is increasing and the intervals on which it is decreasing.

Background

Topic: Increasing/Decreasing Functions

This question tests your ability to use the first derivative to determine where a function is increasing or decreasing.

Key Terms and Formulas:

• First Derivative Test: $f'(x) > 0$ means increasing; $f'(x) < 0$ means decreasing.

Step-by-Step Guidance

1. Rewrite $f(x)$ in simplest form and find $f'(x)$.

2. Solve $f'(x) = 0$ to find critical points.

3. Test intervals between critical points to determine where $f'(x)$ is positive or negative.

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Q11. Use the graphing guidelines to make a complete graph of $f(x) = x^3 - 3x^2 + 2$.

Background

Topic: Curve Sketching

This question tests your ability to analyze a function using calculus (critical points, inflection points, intercepts, end behavior) to sketch its graph.

Key Terms and Formulas:

• Critical Points: Where $f'(x) = 0$

• Inflection Points: Where $f''(x) = 0$

• Intercepts: Where $f(x) = 0$

Step-by-Step Guidance

1. Find $f'(x)$ and $f''(x)$ to locate critical and inflection points.

2. Find $x$- and $y$-intercepts.

3. Analyze end behavior as $x \to \pm\infty$.

4. Combine all information to sketch the graph.

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Q12. Write the equation of the line that represents the linear approximation to the following function at the given point. Use the linear approximation to estimate the given quantity:

$f(x) = \sqrt{x}$ at $x = 25$; estimate $\sqrt{26}$

Background

Topic: Linear Approximation (Tangent Line Approximation)

This question tests your ability to use the tangent line at a point to approximate function values near that point.

Key Terms and Formulas:

• Linear Approximation: $L(x) = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Find $f(a)$ and $f'(a)$ at $a = 25$.

2. Write the linear approximation $L(x)$.

3. Plug $x = 26$ into $L(x)$ to estimate $\sqrt{26}$.

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Q13. Write the equation of the line that represents the linear approximation to the following function at the given point. Use the linear approximation to estimate the given quantity:

$f(x) = \sqrt{x}$ at $x = 25$; estimate $\sqrt{24}$

Background

Topic: Linear Approximation (Tangent Line Approximation)

This question tests your ability to use the tangent line at a point to approximate function values near that point.

Key Terms and Formulas:

• Linear Approximation: $L(x) = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Find $f(a)$ and $f'(a)$ at $a = 25$.

2. Write the linear approximation $L(x)$.

3. Plug $x = 24$ into $L(x)$ to estimate $\sqrt{24}$.

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Q14. A state patrol officer uses a radar gun near a highway on-ramp. The officer could estimate a driver's speed by using the Mean Value Theorem. What can the officer conclude about the speed of the driver?

Background

Topic: Applications of the Mean Value Theorem

This question tests your understanding of how the Mean Value Theorem can be used in real-world scenarios to estimate average and instantaneous rates of change.

Key Terms and Formulas:

• Mean Value Theorem: $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c$ in $(a, b)$

Step-by-Step Guidance

1. Understand that the Mean Value Theorem guarantees the existence of a point where the instantaneous rate equals the average rate over the interval.

2. Relate this to the context of measuring speed over a distance and time interval.

3. Conclude what the officer can infer about the driver's speed at some point.

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Q15. Evaluate the following limit using L'Hôpital's Rule when it is convenient and applicable:

$\lim\limits_{x \to 0} \frac{e^{2x} - e^{-2x}}{x}$

Background

Topic: Limits and L'Hôpital's Rule

This question tests your ability to evaluate limits that result in indeterminate forms and to apply L'Hôpital's Rule.

Key Terms and Formulas:

• L'Hôpital's Rule: If $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ results in $0/0$ or $\infty/\infty$, then $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit on the right exists).

Step-by-Step Guidance

1. Substitute $x = 0$ into the numerator and denominator to check for an indeterminate form.

2. If you get $0/0$, differentiate the numerator and denominator with respect to $x$.

3. Set up the new limit using the derivatives: $\lim\limits_{x \to 0} \frac{f'(x)}{g'(x)}$.

Try solving on your own before revealing the answer!

Q16. Use the graphing guidelines to make a complete graph of $f(x) = x^3 - 3x^2 + 2$.

Background

Topic: Curve Sketching

This question tests your ability to analyze a function using calculus (critical points, inflection points, intercepts, end behavior) to sketch its graph.

Key Terms and Formulas:

• Critical Points: Where $f'(x) = 0$

• Inflection Points: Where $f''(x) = 0$

• Intercepts: Where $f(x) = 0$

Step-by-Step Guidance

1. Find $f'(x)$ and $f''(x)$ to locate critical and inflection points.

2. Find $x$- and $y$-intercepts.

3. Analyze end behavior as $x \to \pm\infty$.

4. Combine all information to sketch the graph.

Try solving on your own before revealing the answer!