Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Evaluate the limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 2

Which of the following statements best describes the meaning of the derivative $$f'(a) at$$ $$x = a$$?

Accepted Answer

It is the slope of the tangent line to $$f(x) at$$ $$x = a.$$

Question 3

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find all critical points.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 4

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 5

Suppose $$f(x) is $$continuous at $$x = 1$$ and $$f(1) = 3. $$Which of the following must be true?

Accepted Answer

$$\lim_{x 	o 1} f(x) = 3$$

Question 6

Find the equation of the tangent line to $$y = x^2$ at x = 1$.

Accepted Answer

$$y = 2x - 1$$

Question 7

If $$f(x) = \sqrt{x}$$, what is $$f'(4)$$?

Accepted Answer

$$\frac{1}{4}$$

Question 8

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 9

A ball is dropped from a height $$h$$ and its height at time $$t is $$given by $$h(t) = h - \frac{1}{2}gt^2$. What is the velocity of the ball at time t$?

Accepted Answer

$$-gt$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 11

If $$f(x) = x^2$ + 2x + 1$$, what is the slope of the tangent line at $$x = -1$$?

Accepted Answer

$$0$$

Question 12

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 13

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x. $$Which statement is true?

Accepted Answer

$$f(x) is $$increasing for all $$x$$

Question 14

Find the derivative of $$f(x) = \frac{1}{x}.$$

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 15

A car's position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is its instantaneous velocity at $$t = 2$$?

Accepted Answer

$$14$$

Question 16

Which of the following is the best interpretation of $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 17

If $$f(x) = x^3$, what is f$$''(x)$$?

Accepted Answer

$$6x$$

Question 18

Which of the following is a necessary condition for a function to have a local maximum at $$x = a$$?

Accepted Answer

$$f'(a) = 0$$

Question 19

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees there is some $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

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