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

$$f'(a) = \lim\limits_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$

Question 1

Given the function $$f(x) = \sqrt{x-5}$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{1}{2\sqrt{x-5}}$$

Question 2

The average rate of change of $$f(x)$$ from $$x=a to$$ $$x=b is $$given by which formula?

Accepted Answer

$$\dfrac{f(b)-f(a)}{b-a}$$

Question 3

Suppose $$f(x) = 4 - x^2$. What is the slope of the tangent line to f$ at the point (0$$,4)$$?

Accepted Answer

Question 4

Given $$f(x) = \sqrt{5x+1}$$, what is the slope of the tangent line at the point $$(3,4)$$?

Accepted Answer

$$\dfrac{5}{8}$$

Question 5

Which of the following statements is TRUE about differentiability and continuity?

Accepted Answer

If a function is differentiable at $$x=a$$, then it is continuous at $$x=a.$$

Question 6

For the function $$f(x) = |x-4|$$, is $$f$$ differentiable at $$x=4$$?

Accepted Answer

No, because the left and right derivatives at $$x=4$$ are not equal.

Question 7

Which of the following is NOT a reason for a function to be non-differentiable at a point $$a$$?

Accepted Answer

The function is linear at $$a.$$

Question 8

Given $$f(x) = \dfrac{1}{3x-1}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -\dfrac{3}{(3x-1)^2$}$$

Question 9

A function $$f is $$continuous at $$x=a$$ but not differentiable at $$x=a. $$Which of the following could be true?

Accepted Answer

The graph of $$f$$ has a sharp corner at $$x=a.$$

Question 10

If $$f(x) is $$differentiable at $$x=a$$, which of the following must be true?

Accepted Answer

$$f(x) is $$continuous at $$x=a.$$

Question 11

Which of the following best describes the instantaneous rate of change of $$f at$$ $$x=a$$?

Accepted Answer

The slope of the tangent line to $$f at$$ $$x=a.$$

Question 12

Given $$f(x) = x^2$, what is the equation of the tangent line at x=1$?

Accepted Answer

Question 13

A function $$f is $$not differentiable at $$x=a if$$:

Accepted Answer

The left-hand and right-hand derivatives at $$x=a$$ are not equal.

Question 14

Suppose the position $$s(t) of a $$car is given by $$s(t) = 3t^2$ + 2t. $$What is the instantaneous velocity at $$t=2$$?

Accepted Answer

$$v(2) = 14$$

Question 15

If $$f(x)$$ has a vertical tangent at $$x=a$$, what can you conclude about $$f'(a)$$?

Accepted Answer

$$f'(a)$$ does not exist.

Question 16

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

Accepted Answer

$$f'(a) = \lim\limits_{h 	o 0} \dfrac{f(a+h) - f(a)}{h}$$

Question 17

A medical researcher models the temperature $$T (in$$ $$^\circC) of a $$patient as $$T(t) = 37 + 0.5t - 0.02t^2$, where t$ is hours after medication. What is the rate of change of temperature at t=5$ hours?

Accepted Answer

$$0.3$$

Question 18

If $$f(x) is $$not continuous at $$x=a$$, which of the following must be true?

Accepted Answer

$$f(x) is $$not differentiable at $$x=a.$$

Question 19

A chemical reaction's concentration is modeled by $$C(t) = 10e$$^{-0.1t}$$. What is the instantaneous rate of change of concentration at t=0$?

Accepted Answer

$$-1$$

Rates of Change, Tangent Lines, and Differentiability in Calculus

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