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

$$\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Question 1

Which of the following best describes the average rate of change of a function $$f on $$the interval $$[a, b]$$?

Accepted Answer

$$\displaystyle \frac{f(b) - f(a)}{b - a}$$

Question 2

The limit definition of the derivative of $$f at$$ $$x = a is$$:

Accepted Answer

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

Question 3

Given $$f(x) = x^2$ - 5x$$, what is the slope of the tangent line to the graph at $$x = 2$$?

Accepted Answer

$$-1$$

Question 4

Which of the following is NOT a correct notation for the derivative of $$f$$?

Accepted Answer

$$f(x)'$$

Question 5

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

Accepted Answer

$$y = 3x - 1$$

Question 6

Using the limit definition, what is $$f'(x)$$ for $$f(x) = 3x^2$ - 2x + 2$$?

Accepted Answer

$$6x - 2$$

Question 7

For $$f(x) = x^2$ - 5x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = -1(x-2) - 6$$

Question 8

Which of the following best describes the geometric meaning of the derivative $$f'(a)$$?

Accepted Answer

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

Question 9

Given $$f(x) = x^2$ + x + 1$$, use the limit definition to find $$f'(x).$$

Accepted Answer

$$2x + 1$$

Question 10

Suppose the position of a car is given by $$s(t) = t^2$ + 2t. $$What does $$s'(t)$$ represent in this context?

Accepted Answer

The instantaneous velocity of the car at time $$t$$

Question 11

If $$f(x) = x^2$ - 5x$$, what is $$f'(x)$$ using the limit definition?

Accepted Answer

$$2x - 5$$

Question 12

Which of the following is the correct process to find the equation of the tangent line to $$f(x) at$$ $$x = a$$?

Accepted Answer

Find $$f(a)$$ and $$f'(a)$$, then use $$y - f(a) = f'(a)(x - a)$$

Question 13

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

Accepted Answer

$$-1$$

Question 14

If $$f(x) = 3x^2$ - 2x + 2$$, what is the value of $$f'(1)$$?

Accepted Answer

$$4$$

Question 15

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

Accepted Answer

$$\displaystyle \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Rates of Change as Limits & The Definition of the Derivative: Differentiability

Study Guide - Practice Questions