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

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

Question 1

A body is dropped from a tall cliff and its distance from the top after $$t$$ seconds is given by $$s = 4.9t^2$. What is the instantaneous velocity of the body at t = 2$ seconds?

Accepted Answer

$$19.6\ \mathrm{m/s}$$

Question 2

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

Accepted Answer

The slope of the tangent to the curve $$y = f(x) at$$ $$x = a$$

Question 3

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 4

If $$f(x) = x^2$ + x$$, what is $$\lim\limits_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 5

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 6

Given $$f(x) = |x|$$ for $$x 
eq 0$$, what is $$\lim\limits_{x 	o 0} f(x)$$?

Accepted Answer

$$0$$

Question 7

For the function $$f(x) = \begin{cases} 1, & x \leq 0 \ 2, & x \u003e 0 \end{cases}$$, what can be said about $$\lim\limits_{x 	o 0} f(x)$$?

Accepted Answer

The limit does not exist

Question 8

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 9

If $$f(x) = x^3$, what is the value of $$\lim\limits_{x 	o 1} f(x)$$?

Accepted Answer

$$1$$

Question 10

Which rule is used to find the derivative of the product of two functions $$u(x)$$ and $$v(x)$$?

Accepted Answer

Product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$

Question 11

Find the derivative of $$f(x) = \sin x.$$

Accepted Answer

$$\cos x$$

Question 12

If $$f(x) = x^n$$ for a positive integer $$n$$, what is $$f'(x)$$?

Accepted Answer

$$nx^{n-1}$$

Question 13

Evaluate $$\lim\limits_{x 	o 0} \frac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 14

Which of the following is true about the limit $$\lim\limits_{x 	o a} f(x)$$ for a polynomial $$f(x)$$?

Accepted Answer

It equals $$f(a)$$

Question 15

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 16

Which of the following is the correct quotient rule for differentiation?

Accepted Answer

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2$}$$

Question 17

Find the derivative of $$f(x) = \cos x.$$

Accepted Answer

$$-\sin x$$

Question 18

If $$f(x) = 3x$$, what is $$f'(2)$$?

Accepted Answer

$$3$$

Question 19

Which mathematicians are credited with the independent invention of calculus?

Accepted Answer

Isaac Newton and G.W. Leibnitz

Limits and Derivatives: Foundations for Physics with Calculus

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