Q: Which of the following best describes the definition of the derivative of a function $$f(x) at a $$point $$x=a$$?

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

Q: Which of the following limits does NOT exist?

$$\displaystyle \lim_{x \to 0} \frac{1}{x}$$

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

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

Question 1

Which of the following best describes the definition of the derivative of a function $$f(x) at a $$point $$x=a$$?

Accepted Answer

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

Question 2

If $$f(x) = x^2$ + 3x$$, what is the equation of the tangent line to $$f(x) at$$ $$x=1$$?

Accepted Answer

$$y = 5x - 1$$

Question 3

Which of the following limits does NOT exist?

Accepted Answer

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

Question 4

A ball is thrown vertically upward with an initial velocity of $$20 m/s. $$Using $$s(t) = -4.9t^2$ + 20t$$, what is the instantaneous velocity at $$t=2$$ seconds?

Accepted Answer

$$0.4 m/s$$

Question 5

Which of the following is the correct derivative of $$f(x) = \sin(x)$$?

Accepted Answer

$$\cos(x)$$

Question 6

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

Accepted Answer

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

Question 7

If $$f(x) = x^3$ - 2x$$, at which $$x-$$value does the function have a horizontal tangent?

Accepted Answer

$$x = 0$$

Question 8

Which of the following statements about the limit $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x-2} is $$true?

Accepted Answer

The limit exists and equals $$4$$

Question 9

A car's position is given by $$s(t) = 3t^2$ + 2t. $$What is the car's acceleration at any time $$t$$?

Accepted Answer

$$6$$

Question 10

Which of the following is the correct application of the product rule for derivatives?

Accepted Answer

$$(fg)' = f'g + fg'$$

Question 11

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 12

Which of the following is the correct chain rule for $$y = f(g(x))$$?

Accepted Answer

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Question 13

A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t=10$?

Accepted Answer

$$100e^{0.5}$$ 	imes 0.05$$

Question 14

Which of the following is the correct derivative of $$f(x) = 	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 15

If $$f(x) = x^2$ and g(x) = x+1$, what is the derivative of f(g(x$$))$$?

Accepted Answer

$$2(x+1)$$

Question 16

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.1t}$$. What is the rate of change of concentration at t=5$?

Accepted Answer

$$-5e^{-0.5}$$

Question 17

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

Accepted Answer

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

Question 18

If $$f(x) = x^4$, what is f$$''(x)$$?

Accepted Answer

$$12x^2$$

Question 19

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 20

A particle moves along a line so that its position at time $$t is$$ $$s(t) = t^3$ - 6t^2$ + 9t. At $$what time(s) is the particle at rest?

Accepted Answer

$$t = 1$$ 	ext{ and } t = 3$

Calculus I Study Notes: Functions, Limits, and Continuity

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