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

Question 1

Which of the following statements is TRUE regarding the relationship between differentiability and continuity at a point $$x=a$$?

Accepted Answer

If $$f is $$differentiable at $$x=a$$, then $$f is $$continuous at $$x=a.$$

Question 2

Given $$f(x) = x^2$ + 3x$$, use the definition of the derivative to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$f'(x) = 2x + 3$$

Question 3

Suppose $$f(x) is $$differentiable at $$x=a. $$Which limit must exist for $$f'(a) to be $$defined?

Accepted Answer

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

Question 4

Which of the following functions is NOT differentiable at $$x=0$$?

Accepted Answer

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

Question 5

If $$C is a $$real number, what is $$\frac{d}{dx}C$$?

Accepted Answer

$$0$$

Question 6

Given $$f(x) = 5x^3$, what is f$$'(x)$$ using the constant multiple rule?

Accepted Answer

$$15x^2$$

Question 7

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

Accepted Answer

$$\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$$

Question 8

Evaluate $$\frac{d}{dx}(2x^3$ + 5x^2$ + 4x + 7).$$

Accepted Answer

$$6x^2 + 10x + 4$

Question 9

Which of the following is a correct statement about the difference rule for differentiation?

Accepted Answer

$$\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$$

Question 10

A car's position is given by $$s(t) = 4t^2$ + 2t + 1. $$What is the car's velocity at time $$t=3$$?

Accepted Answer

$$26$$

Question 11

Which of the following is a valid reason why the sum rule for derivatives cannot be applied to products or quotients?

Accepted Answer

The derivative of a product or quotient involves additional terms not present in the sum rule.

Question 12

Find the value of $$x$$ where the tangent line to $$f(x) = x^3$ - 3x^2$ + 2 is $$horizontal.

Accepted Answer

$$x = 0$$ 	ext{ and } x = 2$

Question 13

If $$f(x) = 7$$, what is $$f'(x)$$?

Accepted Answer

$$0$$

Question 14

Evaluate $$\frac{d}{dx}(x^4$ - 2x^2$ + 3x - 5).$$

Accepted Answer

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

Question 15

Which of the following is the correct derivative of $$f(x) = 2x^3$ - 5x^2$ + 4x + 7$$?

Accepted Answer

$$6x^2 - 10x + 4$

Question 16

If $$f(x) = x^2$ and g(x) = 3x$, what is $$\frac{d}{dx}(f(x) + g(x))$$?

Accepted Answer

$$2x + 3$$

Question 17

Which of the following best describes the geometric meaning of the derivative at a point $$x=a$$?

Accepted Answer

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

Differentiation: Rules, Proofs, and Applications

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