Q: Which of the following is the correct definition of the derivative of a function $$f at$$ $$x = c$$?

$$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$$

Question 1

Which of the following is the correct definition of the derivative of a function $$f at$$ $$x = c$$?

Accepted Answer

$$f'(c) = \lim_{h 	o 0} \frac{f(c+h) - f(c)}{h}$$

Question 2

Given $$f(x) = 7x - 3$$, what is $$f'(x)$$ using the limit definition of the derivative?

Accepted Answer

$$f'(x) = 7$$

Question 3

A ball is thrown straight down from a 220-foot building with an initial velocity of $$-22 ft/s. $$Neglecting air resistance, what is its velocity after 3 seconds? (Use $$v(t) = -32t + v_0$)

Accepted Answer

$$v(3) = -118 ft/s$$

Question 4

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

Accepted Answer

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

Question 5

Find the derivative of $$y = \sqrt{x}.$$

Accepted Answer

$$y' = \frac{1}{2\sqrt{x}}$$

Question 6

If $$y = x^2$ + 2\cos x$$, at which $$x in$$ $$[0, 2\pi]$$ does the graph have a horizontal tangent line?

Accepted Answer

$$x = \arccos(-x/2)$$

Question 7

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

Accepted Answer

$$f'(x) = 4x^3$ + 12x^2$

Question 8

Use the product rule to find the derivative of $$y = (3x - 4)(x^2$ + 5).$$

Accepted Answer

$$y' = (3x - 4) \cdot 2x + (x^2$ + 5) \cdot 3$$

Question 9

Find the derivative of $$y = \frac{1}{x^3$}.$$

Accepted Answer

$$y' = -3x$$^{-4}$$

Question 10

A rock is dropped from a cliff 214 meters above water. The position function is $$s(t) = -4.9t^2$ + 214. $$What is the velocity of the rock at impact?

Accepted Answer

$$v = -64.7 m/s$$

Question 11

Which rule should be used to find the derivative of $$y = x^2$ \sin x$$?

Accepted Answer

Product Rule

Question 12

Find the derivative of $$y = \sin x.$$

Accepted Answer

$$y' = \cos x$$

Question 13

Given $$f(x) = x^2$ + 3$$, find the slope of the tangent line at $$x = 2.$$

Accepted Answer

$$f'(2) = 4$$

Question 14

Find the derivative of $$y = 2x$$^{1/2}$$ + 2\cos x.$$

Accepted Answer

$$y' = x$$^{-1/2}$$ - 2\sin x$$

Question 15

If $$f(x) = x^2$ + 2\cos x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x - 2\sin x$$

Question 16

Find the derivative of $$y = x^2$ + 5x - 3x^2$.

Accepted Answer

$$y' = 5$$

Question 17

Which of the following is the correct application of the quotient rule for $$y = \frac{f(x)}{g(x)}$$?

Accepted Answer

$$y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Question 18

A car moves along a straight line with position $$s(t) = t^3$ + 5t^2$ - 3t + 8. $$What is its velocity function?

Accepted Answer

$$v(t) = 3t^2$ + 10t - 3$$

Question 19

Which theorem states that if $$f is $$differentiable at $$x = c$$, then $$f is $$continuous at $$x = c$$?

Accepted Answer

Differentiability Implies Continuity

Calculus 1 – Chapter 2: Differentiation Study Notes

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