Calculus Test 3 Review – Step-by-Step Guidance

Q1. Find the absolute maximum and minimum values of $f(t) = t + 2\cos t$ on $[0, 2\pi]$.

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the absolute maximum and minimum values of a continuous function on a closed interval using calculus techniques.

Key Terms and Formulas:

• Critical Point: A point where $f'(t) = 0$ or $f'(t)$ does not exist.

• Absolute Maximum/Minimum: The largest/smallest value of $f(t)$ on the interval.

To find absolute extrema on $[a, b]$:

• Find $f'(t)$ and solve $f'(t) = 0$ for critical points in $[a, b]$.

• Evaluate $f(t)$ at endpoints and at each critical point in $[a, b]$.

Step-by-Step Guidance

1. Compute the derivative: $f'(t) = 1 - 2\sin t$.

2. Set $f'(t) = 0$ and solve for $t$ in $[0, 2\pi]$.

3. List all critical points and endpoints: $t = 0$, $t = 2\pi$, and any $t$ found in step 2.

4. Evaluate $f(t)$ at each of these points to compare their values.

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Q2. Verify the function meets the criteria of the Mean Value Theorem (MVT), then find all numbers $c$ that satisfy the conclusion of the MVT for $f(x) = \sqrt{x+1}$ on $[-1, 3]$.

Background

Topic: Mean Value Theorem (MVT)

This question tests your understanding of the conditions for the MVT and your ability to apply it to a specific function and 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 $f(x)$ is continuous on $[-1, 3]$ and differentiable on $(-1, 3)$.

2. Compute $f(-1)$ and $f(3)$.

3. Calculate the average rate of change: $\frac{f(3) - f(-1)}{3 - (-1)}$.

4. Find $f'(x)$ and set $f'(c)$ equal to the average rate of change. Solve for $c$ in $(-1, 3)$.

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Q3. Find the local maximum and minimum values, if they exist, of $h(x) = x^2 \ln x$.

Background

Topic: Local Extrema (Critical Points)

This question tests your ability to find local maxima and minima using the first and second derivative tests.

Key Terms and Formulas:

• Critical Point: Where $h'(x) = 0$ or $h'(x)$ is undefined.

• First Derivative Test: Determines if a critical point is a local max or min.

Step-by-Step Guidance

1. Find the domain of $h(x)$ (where is $\ln x$ defined?).

2. Compute $h'(x)$ using the product rule.

3. Set $h'(x) = 0$ and solve for $x$ to find critical points.

4. Use the first or second derivative test to classify each critical point as a local maximum or minimum.

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Q4. Given $g(x) = x^4 - 2x^2 + 3$, answer the following:

• a. Critical Number(s):

• b. x and y-values of Local minimum(s):

• c. x and y-values of Local Maximum(s):

• d. Open interval(s) where the function increases:

• e. Open interval(s) where the function decreases:

• f. x and y-values of Inflection point(s):

• g. Open interval(s) where the function is concave up:

• h. Open interval(s) where the function is concave down:

Background

Topic: Curve Sketching (Critical Points, Intervals of Increase/Decrease, Concavity, Inflection Points)

This question tests your ability to analyze a function using its first and second derivatives.

Key Terms and Formulas:

• Critical Number: Where $g'(x) = 0$ or $g'(x)$ is undefined.

• Inflection Point: Where $g''(x) = 0$ and concavity changes.

• Increasing/Decreasing: Determined by the sign of $g'(x)$.

• Concave Up/Down: Determined by the sign of $g''(x)$.

Step-by-Step Guidance

1. Compute $g'(x)$ and solve $g'(x) = 0$ for critical numbers.

2. Use the first derivative test to determine intervals of increase/decrease and classify local extrema.

3. Compute $g''(x)$ and solve $g''(x) = 0$ for possible inflection points.

4. Use the second derivative test to determine concavity and confirm inflection points.

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Q5. Evaluate the limit $\lim_{t \to 0} t \csc t$.

Background

Topic: Limits Involving Trigonometric Functions

This question tests your understanding of limits involving trigonometric functions, especially as the variable approaches zero.

Key Terms and Formulas:

• $\csc t = \frac{1}{\sin t}$

• Standard limit: $\lim_{t \to 0} \frac{\sin t}{t} = 1$

Step-by-Step Guidance

1. Rewrite $t \csc t$ as $\frac{t}{\sin t}$.

2. Recognize the standard limit form and recall the relevant limit property.

3. Apply the limit property to evaluate the expression as $t \to 0$.

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Q6. Evaluate the limit $\lim_{x \to \infty} e^{\frac{1}{x} - 1} \ln x$.

Background

Topic: Limits at Infinity, Exponential and Logarithmic Functions

This question tests your ability to analyze limits involving exponential and logarithmic functions as $x$ approaches infinity.

Key Terms and Formulas:

• As $x \to \infty$, $\frac{1}{x} \to 0$.

• Recall properties of $e^x$ and $\ln x$ as $x \to \infty$.

Step-by-Step Guidance

1. Analyze the exponent: $\frac{1}{x} - 1$ as $x \to \infty$.

2. Determine the behavior of $e^{\frac{1}{x} - 1}$ as $x \to \infty$.

3. Examine the behavior of $\ln x$ as $x \to \infty$.

4. Combine the results to analyze the overall limit.

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Q7. Evaluate the limit $\lim_{x \to -\infty} x^2 e^{2x}$.

Background

Topic: Limits at Infinity, Exponential Dominance

This question tests your understanding of how exponential functions compare to polynomial functions as $x$ approaches negative infinity.

Key Terms and Formulas:

• As $x \to -\infty$, $e^{2x} \to 0$ very rapidly.

• Compare the rates at which $x^2$ grows and $e^{2x}$ decays.

Step-by-Step Guidance

1. Analyze the behavior of $x^2$ as $x \to -\infty$.

2. Analyze the behavior of $e^{2x}$ as $x \to -\infty$.

3. Consider the product $x^2 e^{2x}$ and which term dominates as $x \to -\infty$.

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Q8. Evaluate the limit $\lim_{x \to 0} x^x$.

Background

Topic: Indeterminate Forms, Exponential Limits

This question tests your ability to evaluate limits of the form $0^0$, which is an indeterminate form, often using logarithms and L'Hospital's Rule.

Key Terms and Formulas:

• Rewrite $x^x$ as $e^{x \ln x}$.

• Analyze the limit of $x \ln x$ as $x \to 0^+$.

Step-by-Step Guidance

1. Let $y = x^x$, then take the natural logarithm: $\ln y = x \ln x$.

2. Analyze $\lim_{x \to 0^+} x \ln x$ (use substitution if needed).

3. Exponentiate the result to find the limit of $x^x$.

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Q9. A jet flying at 600 ft/sec and travelling in a straight line at a constant elevation of 900 ft passes directly over a spectator at an airshow. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?

Background

Topic: Related Rates (Trigonometric Applications)

This question tests your ability to set up and solve a related rates problem involving trigonometric relationships.

Key Terms and Formulas:

• Let $x$ be the horizontal distance from the spectator to the jet, $h = 900$ ft, $\theta$ the angle of elevation.

• $\tan \theta = \frac{h}{x}$

• Given $\frac{dx}{dt} = 600$ ft/sec.

Step-by-Step Guidance

1. Express $\tan \theta$ in terms of $x$ and $h$.

2. Differentiate both sides with respect to $t$ to relate $\frac{d\theta}{dt}$ and $\frac{dx}{dt}$.

3. Find $x$ at $t = 2$ seconds (since $x = 600t$).

4. Plug in all known values to solve for $\frac{d\theta}{dt}$ at $t = 2$.

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Q10. Use linear approximations to estimate the following quantities:

• a. $\sqrt{35}$

• b. $\pi^2$

Background

Topic: Linear Approximation (Tangent Line Approximation)

This question tests your ability to use the tangent line at a known point to estimate the value of a function near that point.

Key Terms and Formulas:

• Linear approximation: $f(x) \approx f(a) + f'(a)(x - a)$ for $x$ near $a$.

Step-by-Step Guidance

1. Choose a value $a$ near the value you want to estimate, where $f(a)$ and $f'(a)$ are easy to compute.

2. Compute $f(a)$ and $f'(a)$ for each function.

3. Plug $x$ and $a$ into the linear approximation formula to estimate the desired value.

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Q11. Suppose you had 102 m of fencing to make two side-by-side enclosures as shown. What is the maximum area that you could enclose?

Background

Topic: Optimization (Applications of Derivatives)

This question tests your ability to set up and solve an optimization problem involving constraints and maximizing area.

Key Terms and Formulas:

• Let $x$ and $y$ be the dimensions of the enclosure.

• Write equations for the total fencing used and the area enclosed.

• Express area as a function of one variable using the constraint.

Step-by-Step Guidance

1. Draw a diagram and label the dimensions. Write the equation for the total fencing used.

2. Express the area in terms of $x$ and $y$.

3. Use the constraint to write area as a function of one variable.

4. Take the derivative, set it to zero, and solve for the critical point(s) to maximize area.

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