Question 1

Suppose the radius $r$ of a sphere is measured as $5$ cm with a possible error of $0.1$ cm. Using differentials, estimate the maximum error in the computed volume of the sphere.

Accepted Answer

$dV = 4\pi (5)^2 (0.1) = 10\pi$

Question 2

Given $f(x) = x^3 + x^2$, if $x = 2$ and $dx = 0.01$, what is the approximate change $df$ in $f(x)$?

Accepted Answer

$df = f'(2) dx = (3(2)^2 + 2(2)) (0.01) = (12 + 4)(0.01) = 0.16$

Question 3

A cube has its side measured as $6$ cm with an error of $0.2$ cm. Using differentials, estimate the error in the computed volume.

Accepted Answer

$dV = 3x^2 dx = 3(6)^2 (0.2) = 21.6$

Question 4

Which of the following statements about absolute maximum and minimum values is correct?

Accepted Answer

An absolute maximum of $f$ on $D$ is $f(c)$ if $f(c) \geq f(x)$ for all $x$ in $D$.

Question 5

For the function $f(x) = x^3$, which of the following is true about its absolute extrema on $(-\infty, \infty)$?

Accepted Answer

It has no absolute maximum or minimum.

Question 6

Consider the function $f(x) = x^2$ on the domain $(-\infty, \infty)$. What is true about its absolute extrema?

Accepted Answer

It has an absolute minimum at $x = 0$ and no absolute maximum.

Question 7

Which theorem guarantees that a continuous function on a closed interval $[a, b]$ attains both an absolute maximum and minimum?

Accepted Answer

Extreme Value Theorem

Question 8

Let $f(x)$ be continuous on $[a, b]$. Which of the following must be true?

Accepted Answer

$f(x)$ attains both an absolute maximum and minimum on $[a, b]$.

Question 9

According to Fermat's Theorem, if $f$ has a local maximum or minimum at $x = c$ and $f$ is differentiable at $c$, then:

Accepted Answer

$f'(c) = 0$

Question 10

Which of the following is a critical point of $f(x)$?

Accepted Answer

Any $x$ in the domain of $f$ where $f'(x) = 0$ or $f'(x)$ is undefined.

Question 11

Find all critical points of $f(x) = x^2 - x$.

Accepted Answer

$x = 0$ and $x = 1$

Question 12

Given $g(y) = \dfrac{y^2 - y + 1}{y^2 - y + 1}$, what are its critical numbers?

Accepted Answer

$y = 0$ and $y = 2$

Question 13

A medical researcher measures the diameter of a blood vessel as $2$ mm with a possible error of $0.05$ mm. Estimate the error in the computed cross-sectional area using differentials.

Accepted Answer

$dA = 2\pi r dr = 2\pi (2)(0.05) = 0.2\pi$

Question 14

Which of the following intervals for $f(x) = x$ on $[0,1]$ contains an absolute maximum?

Accepted Answer

$[0,1]$

Question 15

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, which theorem ensures that $f(x)$ attains its absolute extrema on $[a, b]$?

Accepted Answer

Extreme Value Theorem

Question 16

Which of the following is NOT a step in finding critical points of a function $f(x)$?

Accepted Answer

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

Question 17

A population biologist measures the radius of a circular pond as $10$ m with a possible error of $0.1$ m. Estimate the error in the computed area using differentials.

Accepted Answer

$dA = 2\pi r dr = 2\pi (10)(0.1) = 2\pi$

Question 18

Which of the following best describes a local maximum of $f(x)$ at $x = c$?

Accepted Answer

$f(c) \geq f(x)$ for $x$ near $c$

Applications of Differentiation: Error Propagation, Maximum and Minimum Values, and Critical Points

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