Question 1

Suppose $$f(x) is $$continuous at $$x = 2$$ but not differentiable at $$x = 2. $$Which of the following could be true?

Accepted Answer

$$f(x)$$ has a sharp corner at $$x = 2$$

Question 2

Given $$f(x) = x^3$ + 5x^2$ + 10x + 5$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 + 10x + 10$

Question 3

The average rate of change of a function $$f(x)$$ from $$x = 0 to$$ $$x = 1 is $$given by:

Accepted Answer

$$\dfrac{f(1) - f(0)}{1 - 0}$$

Question 4

If $$f(x) = e$$^{-\cos(x)}$$, what is f$$'(x)$$?

Accepted Answer

$$e^{-\cos(x)}$$ \sin(x)$$

Question 5

A medication's concentration in the bloodstream is modeled by $$C(t) = \sqrt{t} - t^2$. What is the instantaneous rate of change of concentration at t = 4$ hours?

Accepted Answer

$$\dfrac{1}{2\sqrt{4}} - 2 \cdot 4 = \dfrac{1}{4} - 8 = -\dfrac{31}{4}$$

Question 6

Which of the following statements about the relationship between a function and its derivative is correct?

Accepted Answer

The graph of $$f'(x)$$ gives the slope of the tangent line to $$f(x) at $$each point.

Question 7

Find the derivative of $$g(x) = \dfrac{x}{x-3}.$$

Accepted Answer

$$\dfrac{-3}{(x-3)^2$}$$

Question 8

A population of bacteria grows according to $$P(t) = 3e$$^{2t}$$. What is the rate of growth at t = 1$ hour?

Accepted Answer

$$6e^{2}$$

Question 9

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 10

Given $$h(t) = t$$^{3/2}$$ - 5t$$^{1/2}$$, what is h$$'(t)$$?

Accepted Answer

$$\dfrac{3}{2} t$$^{1/2}$$ - \dfrac{5}{2} t$$^{-1/2}$$

Question 11

A car's position is given by $$s(t) = t^3$ - t. $$What is the velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$11$$

Question 12

If $$f(x) is $$continuous at $$x = 0$$ and $$\lim_{x 	o 0} f(x) = 2$$, which of the following must be true?

Accepted Answer

$$f(0) = 2$$

Question 13

Find the derivative of $$f(t) = (5 - t^4$)^5 (5$$ - $$4t^3$$)$.

Accepted Answer

Question 14

If $$f(x) = e$$^{1-x}$$, what is f$$'(x)$$?

Accepted Answer

$$-e^{1-x}$$

Question 15

Suppose $$y = 4t + 3 is $$the equation of a tangent line to $$f(t) at$$ $$t = 1. $$What is $$f(1)$$?

Accepted Answer

$$7$$

Question 16

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 17

Given $$f(x) = [\sin(x^2$)]^5$$, what is $$f'(x)$$?

Accepted Answer

$$5 [\sin(x^2$)]^4 \cos(x^2$) \cdot 2x$$

Question 18

A function $$f(x)$$ has $$\lim_{x 	o 2^-} f(x) = 4$$, $$\lim_{x 	o 2^+} f(x) = 0$$, and $$f(2) = 4. $$Which of the following is true?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2$$

Question 19

The instantaneous rate of change of a function at a point is given by:

Accepted Answer

The slope of the tangent line at that point

Question 20

If $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 0 \ 2x, & x \geq 0 \end{cases}$$, is $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

No, because the left and right derivatives at $$x = 0$$ are not equal

Calculus 1A: Limits, Derivatives, and Rates of Change – Study Notes

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