$$f'(x_0$) = \lim_{h \to 0} \dfrac{f(x_0$ + h) - f(x_0$)}{h}$$

$$\lim_{h \to 0^+} \dfrac{f(a + h) - f(a)}{h}$$ and $$\lim_{h \to 0^-} \dfrac{f(b + h) - f(b)}{h}$$

Question 1

Which of the following best describes the geometric meaning of the derivative $$f'(x_0$) at a $$point $$x_0$$ for a function $$y = f(x)$$?

Accepted Answer

It is the slope of the tangent line to the graph of $$y = f(x) at$$ $$x_0.$$

Question 2

Given the definition of the derivative at a point, which of the following is the correct formula for $$f'(x_0$)$$?

Accepted Answer

$$f'(x_0$) = \lim_{h 	o 0} \dfrac{f(x_0$ + h) - f(x_0$)}{h}$$

Question 3

Suppose $$y = \dfrac{1}{x}$$ for $$x \u003e 0. $$What is the slope of the tangent line at $$x = 2$$?

Accepted Answer

$$-\dfrac{1}{4}$$

Question 4

For the function $$f(x) = \sqrt{x}$$, which of the following is the correct expression for its derivative for $$x \u003e 0$$?

Accepted Answer

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

Question 5

Which of the following statements is TRUE about differentiability and continuity at a point $$x = c$$?

Accepted Answer

If $$f is $$differentiable at $$x = c$$, then $$f$$ must be continuous at $$x = c.$$

Question 6

Which of the following is NOT a reason why a function might fail to have a derivative at a point?

Accepted Answer

The function is linear at the point.

Question 7

Consider the function $$f(x) = x^n$$ where $$n is a $$positive integer. What is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = n x$$^{n-1}$$

Question 8

If $$f(x) is a $$constant function, what is its derivative with respect to $$x$$?

Accepted Answer

$$0$$

Question 9

Given $$f(x) = x^3$ - 2x^2$ + 4x - 5$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 - 4x + 4$

Question 10

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

Accepted Answer

$$\dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 11

A function $$f(x) is $$differentiable on the open interval $$(a, b)$$ and at the endpoints $$a$$ and $$b if $$which of the following limits exist?

Accepted Answer

$$\lim_{h 	o 0^+} \dfrac{f(a + h) - f(a)}{h}$$ and $$\lim_{h 	o 0^-} \dfrac{f(b + h) - f(b)}{h}$$

Question 12

For the function $$f(x) = \sqrt{x}$$, why does the derivative not exist at $$x = 0$$?

Accepted Answer

Because the right-hand limit of $$\dfrac{1}{2\sqrt{x}} as$$ $$x 	o 0^+ is $$infinite.

Question 13

Which of the following functions has a point where the derivative does NOT exist?

Accepted Answer

$$f(x) = |x|$$

Question 14

A function $$y = f(x)$$ has a cusp at $$x = c. $$What does this mean for the derivative at $$x = c$$?

Accepted Answer

The derivative does not exist at $$x = c$$ because the slope approaches $$+\infty$$ from one side and $$-\infty$$ from the other.

Question 15

Which of the following is an example of a function whose derivative fails to exist due to rapid oscillation near a point?

Accepted Answer

$$f(x) = \sin\left(\dfrac{1}{x}\right) at$$ $$x = 0$$

Question 16

If $$f(x) = 3x^2$, what is the effect on the derivative if we multiply the function by a constant c$?

Accepted Answer

The derivative is multiplied by $$c$$; $$\dfrac{d}{dx}[c f(x)] = c f'(x).$$

Question 17

Which of the following is the correct difference quotient for the secant slope between points $$x$$ and $$z on $$the graph of $$y = f(x)$$?

Accepted Answer

$$\dfrac{f(z) - f(x)}{z - x}$$

Question 18

Which of the following is NOT an interpretation of the limit $$\lim_{h 	o 0} \dfrac{f(x_0$ + h) - f(x_0$)}{h}$$?

Accepted Answer

The area under the curve $$y = f(x)$$ from $$x_0 to$$ $$x_0 + h$.

Derivatives: Tangents, Rates of Change, and Basic Rules

스터디 가이드 - 연습 문제