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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