Question 1

Which of the following statements best describes the Extreme Value Theorem (EVT)?

Accepted Answer

If $$f is $$continuous on a closed interval $$[a, b]$$, then $$f$$ has at least one absolute maximum and one absolute minimum on $$[a, b].$$

Question 2

Given $$f(x) = 2x^2$ - 3x + 1 on$$ $$[0, 2]$$, which value(s) of $$c$$ satisfy the conclusion of the Mean Value Theorem?

Accepted Answer

$$c = 0.25$$

Question 3

A farmer wants to fence in an area of $$1.5$$ million square feet in a rectangular field and divide it in half with a fence parallel to one side. What is the minimum total length of fencing required?

Accepted Answer

The minimum occurs when the dividing fence is parallel to the longer side, and the field is twice as long as it is wide.

Question 4

Suppose $$f(x) = x^2$ - 4 on$$ $$[0, 3]. $$What does the midpoint Riemann sum with $$n = 6$$ represent?

Accepted Answer

An approximation of the area under $$f(x)$$ from $$x = 0 to$$ $$x = 3$$ using the midpoints of each subinterval.

Question 5

Which of the following is the correct formula for the average value of a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

$$f_{ave} = \frac{1}{b-a} \int_a^b f(x) dx$$

Question 6

For the function $$f(x) = \frac{x}{x-1}$$, which of the following statements is true about its symmetry?

Accepted Answer

$$f(x)$$ has no symmetry

Question 7

Given $$f(x) = x^2$ - 4$$, what is the value of the definite integral $$\$$int_0$$^3 f(x) dx$$?

Accepted Answer

$$-3$$

Question 8

Which of the following best describes the use of L'Hospital's Rule?

Accepted Answer

It is used to evaluate limits that result in indeterminate forms like $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}.$$

Question 9

If $$f''(x) \u003e 0 on an $$interval, what can you conclude about the graph of $$f on $$that interval?

Accepted Answer

The graph is concave up.

Question 10

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, what must be true?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 11

Given the function $$f(x) = x^2$ - 4 on$$ $$[0, 3]$$, and using a midpoint Riemann sum with $$n = 6$$, what is the width $$\Delta x of $$each subinterval?

Accepted Answer

$$\Delta x = 0.5$$

Question 12

Which of the following is the correct linear approximation (linearization) of $$f(x) = \sqrt{x+3} at$$ $$a = 1$$?

Accepted Answer

$$L(x) = 2 + \frac{1}{4}(x-1)$$

Question 13

For the function $$f(x) = x^2$ - 4$$, what is the average value of $$f on$$ $$[0, 3]$$?

Accepted Answer

$$-1$$

Question 14

Which of the following is a necessary condition for a point $$c to be a $$critical point of $$f$$?

Accepted Answer

$$f'(c) = 0 or$$ $$f'(c)$$ does not exist

Question 15

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f(a) = f(b)$$, what does Rolle's Theorem guarantee?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f'(c) = 0$$

Question 16

If $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude?

Accepted Answer

$$f(x) is $$increasing on that interval.

Question 17

Which of the following is the correct application of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $$g(x) = \int_a^x f(t) dt$$, then $$g'(x) = f(x).$$

Mean Value Theorem and Curve Analysis – Step-by-Step Calculus Guidance

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