Q: What is the limit definition of the derivative of $$f(x) at$$ $$x_0$$?

$$\displaystyle \lim_{h \to 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x_0$$?

Accepted Answer

The slope of the tangent line to the curve $$y = f(x) at$$ $$x = x_0$.

Question 2

What is the limit definition of the derivative of $$f(x) at$$ $$x_0$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$

Question 3

If $$f(x) = \frac{1}{x}$$, what is the slope of the tangent line at $$x = 2$$?

Accepted Answer

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

Question 4

Which of the following is NOT an interpretation of the derivative $$f'(x_0$)$$?

Accepted Answer

The value of $$f(x_0$)$$ itself.

Question 5

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

Accepted Answer

$$2$$

Question 6

The tangent slopes for $$f(x) = \frac{1}{x}$$ become more gradual as the point of tangency moves away from the origin. What does this imply about $$f'(x) as$$ $$|x|$$ increases?

Accepted Answer

$$f'(x)$$ approaches $$0 as$$ $$|x|$$ increases.

Question 7

Which of the following statements is TRUE about the tangent line to the curve $$y = f(x) at$$ $$x = x_0$?

Accepted Answer

It passes through the point $$(x_0$, f(x_0$))$$ and has slope $$f'(x_0$).$$

Question 8

Suppose $$f(x)$$ models the concentration of a drug in the bloodstream over time. What does $$f'(t_0$)$$ represent in this context?

Accepted Answer

The rate at which the drug concentration is changing at time $$t_0.$$

Question 9

If the derivative $$f'(x_0$)$$ does not exist, what can be said about the curve $$y = f(x) at$$ $$x = x_0$?

Accepted Answer

The curve may have a sharp corner or cusp at $$x_0.$$

Question 10

Given $$f(x) = x^3$, use the limit definition to find f$$'(x)$$.

Accepted Answer

$$3x^2$$

Question 11

Which of the following real-world scenarios best illustrates the concept of a derivative?

Accepted Answer

The speedometer in a car showing the instantaneous speed.

Question 12

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

Accepted Answer

$$0$$

Question 13

Which of the following is the correct interpretation of $$\displaystyle \lim_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x_0.$$

Question 14

For $$f(x) = \frac{1}{x}$$, what happens to the slope of the tangent line as $$x$$ approaches $$0$$?

Accepted Answer

The slope becomes infinitely steep.

Section 3.1: Tangent and the Derivative at a Point

Study Guide - Practice Questions

Study Guide - Flashcards