Q: Let $$f(x) = 3x^2$ - 8. $$Using the limit definition, compute $$f'(2). $$Show your steps.

$$f'(2) = \lim_{h \to 0} \frac{3(2+h)^2$ - 8 - [3(4) - 8]}{h} = 12$$

Question 1

Let $$f(x) = 3x^2$ - 8. $$What is the derivative $$f'(x)$$ using the definition of the derivative?

Accepted Answer

$$f'(x) = 6x$$

Question 2

Suppose $$f(x) is $$not continuous at $$x = a. $$What can you conclude about the differentiability of $$f at$$ $$x = a$$?

Accepted Answer

$$f is $$not differentiable at $$x = a$$

Question 3

Given $$f(x) = |x|$$, which of the following statements is true about $$f'(0)$$?

Accepted Answer

$$f'(0)$$ does not exist because there is a corner at $$x = 0$$

Question 4

If $$f(x)$$ has a vertical tangent at $$x = a$$, what happens to $$f'(a)$$?

Accepted Answer

$$f'(a)$$ does not exist (DNE)

Question 5

A function $$f(x) is $$defined as $$f(x) = x^3$ + 8$$ for $$x \u003e 0. $$What is the slope of the tangent line to $$f at$$ $$x = 2$$?

Accepted Answer

$$f'(2) = 12$$

Question 6

Which of the following is NOT a reason for a function to be non-differentiable at $$x = a$$?

Accepted Answer

The function is constant at $$x = a$$

Question 7

Given $$f(x) = 3x^2$ - 8$$, find the equation of the tangent line to $$f at$$ $$x = 1.$$

Accepted Answer

$$y = 6x - 11$$

Question 8

If $$f(x) is $$increasing on an interval, what can you say about $$f'(x) on $$that interval?

Accepted Answer

$$f'(x) \u003e 0$$

Question 9

Suppose $$f(x) is $$differentiable everywhere except at $$x = 0$$, where $$f(x)$$ has a hole. What is true about $$f'(0)$$?

Accepted Answer

$$f'(0)$$ does not exist$

Question 10

Let $$f(x) = 3x^2$ - 8. $$Using the limit definition, compute $$f'(2). $$Show your steps.

Accepted Answer

$$f'(2) = \lim_{h 	o 0} \frac{3(2+h)^2$ - 8 - [3(4) - 8]}{h} = 12$$

Question 11

A car's position is given by $$s(t) = 3t^2$ - 8$$ meters after $$t$$ seconds. What is the car's instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$18 m/s$$

Question 12

If $$f(x) is $$decreasing on $$(-\infty, 0)$$ and increasing on $$(0, \infty)$$, what can you say about $$f'(0) if$$ $$f is $$differentiable at $$x = 0$$?

Accepted Answer

$$f'(0) = 0$$

Question 13

Given $$f(x) = |x|$$, which of the following best describes the graph of $$f'(x)$$?

Accepted Answer

$$f'(x) = 1$$ for $$x \u003e 0$$, $$f'(x) = -1$$ for $$x \u003c 0$$, and $$f'(0)$$ does not exist

Question 14

Suppose $$f(x) is $$defined as $$f(x) = 3x^2$ - 8$$ for $$x \geq 0$$ and $$f(x) = -3x^2$ + 8$$ for $$x \u003c 0. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

No, because the left and right derivatives at $$x = 0$$ are not equal

Question 15

If $$f(x)$$ has a horizontal tangent at $$x = a$$, what does this imply about $$f'(a)$$?

Accepted Answer

$$f'(a) = 0$$

Question 16

A function $$f(x) is $$defined as $$f(x) = x$$^{1/3}$$. What is true about the differentiability of f$ at x = 0$?

Accepted Answer

$$f'(0)$$ exists and is finite

Question 17

Let $$f(x) be a $$function with $$f'(x) = 0$$ for all $$x in an $$interval. What can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$constant on that interval

Question 18

Suppose $$f(x) is $$defined as $$f(x) = 3x^2$ - 8$$ for $$x \u003e 0$$ and $$f(x) = 0$$ for $$x \leq 0. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

No, because the left and right derivatives at $$x = 0$$ are not equal

Question 19

A chemical reaction's temperature (in $$^\circC) is $$modeled by $$T(t) = 3t^2$ - 8$$ where $$t is $$time in minutes. How fast is the temperature changing at $$t = 5$$ minutes?

Accepted Answer

$$30$$ $$^\circC/$$min

Derivatives and Differentiability: Concepts, Computation, and Graphical Interpretation

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