Calculus I Unit 4 Review – Step-by-Step Study Guidance

Q1. Estimate $\sqrt{101}$ using linear approximation and use the calculator to estimate the error.

Background

Topic: Linear Approximation (Linearization)

This question tests your ability to use the tangent line (linearization) to approximate the value of a function near a known point, and then compare your estimate to the actual value.

Key Terms and Formulas

• Linearization of $f(x)$ at $x=a$: $L(x) = f(a) + f'(a)(x - a)$

• For $f(x) = \sqrt{x}$, $f'(x) = \frac{1}{2\sqrt{x}}$

Step-by-Step Guidance

1. Choose a value of $a$ close to $101$where$\sqrt{a}$is easy to compute. (Hint:$a=100$ is a good choice.)

2. Compute $f(a) = \sqrt{a}$ and $f'(a) = \frac{1}{2\sqrt{a}}$.

3. Write the linearization formula: $L(x) = f(a) + f'(a)(x - a)$.

4. Plug in $x = 101$ into your linearization to estimate $\sqrt{101}$.

5. Use a calculator to find the actual value of $\sqrt{101}$ and compare it to your estimate to find the error.

Try solving on your own before revealing the answer!

Q2. Use linear approximation to estimate $\Delta f$ for $f(625.14) - f(624.95)$ (rounded to five decimal places). Then, find the percent error of the estimate.

Background

Topic: Linear Approximation and Error Analysis

This question asks you to use the linear approximation to estimate the change in a function's value over a small interval, and then assess the accuracy of your estimate.

Key Terms and Formulas

• Linear approximation: $\Delta f \approx f'(a) \Delta x$ where $\Delta x = x - a$

• Percent error: $\frac{|\text{estimate} - \text{actual}|}{|\text{actual}|} \times 100\%$

Step-by-Step Guidance

1. Identify the function $f(x)$ (if not given, assume $f(x) = \sqrt{x}$ or as specified in your class).

2. Choose a value $a$ close to both $625.14$ and $624.95$ (e.g., $a = 625$).

3. Compute $f'(a)$.

4. Calculate $\Delta x = 625.14 - 624.95$.

5. Use the linear approximation formula to estimate $\Delta f$.

6. Find the actual $\Delta f$ using a calculator, then compute the percent error.

Try solving on your own before revealing the answer!

Q3a. Find the linearization of $v(t) = 32t - 4t^2$ at $a = 2$.

Background

Topic: Linearization of a Function at a Point

This question asks you to find the equation of the tangent line (linearization) to a quadratic function at a specific point.

Key Terms and Formulas

• Linearization: $L(t) = f(a) + f'(a)(t - a)$

• Derivative: $f'(t) = \frac{d}{dt} [32t - 4t^2]$

Step-by-Step Guidance

1. Compute $f(a)$ by plugging $t = 2$ into $v(t)$.

2. Find $f'(t)$ by differentiating $v(t)$ with respect to $t$.

3. Evaluate $f'(a)$ at $t = 2$.

4. Write the linearization formula $L(t) = f(a) + f'(a)(t - a)$ using your computed values.

Try solving on your own before revealing the answer!

Q3b. Find the linearization of $f(x) = \ln(x + e)$ at $a = e$.

Background

Topic: Linearization of a Logarithmic Function

This question asks you to find the tangent line approximation for a logarithmic function at a specific point.

Key Terms and Formulas

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

• Derivative: $f'(x) = \frac{1}{x + e}$

Step-by-Step Guidance

1. Compute $f(a)$ by plugging $x = e$ into $f(x)$.

2. Find $f'(x)$ and evaluate at $x = e$.

3. Write the linearization formula $L(x) = f(a) + f'(a)(x - a)$ using your computed values.

Try solving on your own before revealing the answer!

Q4. For $f(x) = 2x^3 - 3x^2 - 36x + 12$, find any critical points, extreme values, and intervals where the function is increasing or decreasing.

Background

Topic: Critical Points and Increasing/Decreasing Intervals

This question tests your ability to use the first derivative to find where a function has critical points, local extrema, and to determine where it is increasing or decreasing.

Key Terms and Formulas

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

• First derivative: $f'(x)$

• Test intervals around critical points to determine increasing/decreasing behavior

Step-by-Step Guidance

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

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

3. Set up intervals between critical points and test the sign of $f'(x)$ in each interval.

4. Determine where $f(x)$ is increasing (where $f'(x) > 0$) and decreasing (where $f'(x) < 0$).

5. Use the first derivative test to identify local maxima and minima at the critical points.

Try solving on your own before revealing the answer!

Q5. For $s(t) = t^2 - t$, find any critical points, extreme values, and intervals where the function is increasing or decreasing.

Background

Topic: Critical Points and Increasing/Decreasing Intervals

This question is similar to Q4 but with a quadratic function.

Key Terms and Formulas

• Critical points: where $s'(t) = 0$ or $s'(t)$ is undefined

• First derivative: $s'(t)$

Step-by-Step Guidance

1. Find $s'(t)$ by differentiating $s(t)$.

2. Solve $s'(t) = 0$ to find critical points.

3. Test intervals around the critical point to determine increasing/decreasing behavior.

4. Use the first derivative test to identify local extrema.

Try solving on your own before revealing the answer!

Q6. For $g(x) = \sin(2x) + 3$ on $[-\pi, \pi]$, find any critical points, extreme values, and intervals where the function is increasing or decreasing.

Background

Topic: Trigonometric Functions – Critical Points and Extrema

This question involves finding where a trigonometric function is increasing or decreasing and identifying its critical points and extrema on a closed interval.

Key Terms and Formulas

• Critical points: where $g'(x) = 0$ or $g'(x)$ is undefined

• First derivative: $g'(x) = 2\cos(2x)$

Step-by-Step Guidance

1. Find $g'(x)$ by differentiating $g(x)$.

2. Solve $g'(x) = 0$ for $x$ in $[-\pi, \pi]$ to find critical points.

3. Test intervals between critical points to determine where $g(x)$ is increasing or decreasing.

4. Evaluate $g(x)$ at critical points and endpoints to find extreme values.

Try solving on your own before revealing the answer!

Q7. For $f(x) = x^9 - x$, find any critical points, extreme values, and intervals where the function is increasing or decreasing.

Background

Topic: Polynomial Functions – Critical Points and Extrema

This question involves a higher-degree polynomial and asks for the same analysis as previous questions.

Key Terms and Formulas

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

• First derivative: $f'(x) = 9x^8 - 1$

Step-by-Step Guidance

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

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

3. Test intervals around critical points to determine increasing/decreasing behavior.

4. Use the first derivative test to identify local extrema.

Try solving on your own before revealing the answer!