Question 1

Given the function $$f(x) = x^5$ - 3x^4$ + x$$, what is the third derivative $$f^{(3)}(x$$)$$?

Accepted Answer

$$60x^2 - 72x$

Question 2

Which of the following best describes the physical meaning of the second derivative of a position function $$s(t)$$?

Accepted Answer

It represents the acceleration of the object.

Question 3

If $$y = \frac{1}{x}$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$2x^{-3} or$$ $$\frac{2}{x^3$}$$

Question 4

For $$y = (x^2$ + 10x$$)^{20}$$, what is y$$'$$ using the chain rule?

Accepted Answer

$$20(x^2$$ + $$10x)^{19}(2x + 10$$)$$

Question 5

Given $$s(t) = 10t^2$, where s$ is in miles and t$ in hours, what is the velocity v(4$$)$$ when t = 4$ hours?

Accepted Answer

$$80 mi/hr$$

Question 6

What is the fourth derivative $$\frac{d^4$}{dx^4$}\left[\frac{1}{x}\right]$$?

Accepted Answer

$$\frac{24}{x^5$}$$

Question 7

A pebble is dropped from a hot-air balloon. If $$s(t) = 16t^2$, what is its acceleration after 3.5$ seconds?

Accepted Answer

$$32$$ 	ext{ feet/second}^2$

Question 8

Which notation correctly represents the $$nth $$derivative of a function $$f(x)$$?

Accepted Answer

$$f^{(n)}(x$$)$$ or \frac{d^n}{dx^n}f(x)$

Question 9

If $$y = -6x^4$ + 3x^2$, what is y$$''$$?

Accepted Answer

$$-72x^2 + 6$

Question 10

Which of the following is NOT a correct interpretation of the second derivative $$f''(x)$$?

Accepted Answer

It represents the total area under the curve $$f(x).$$

Question 11

Given $$f(x) = x^5$ - 3x^4$ + x$$, what is $$f^{(6)}(x$$)$$?

Accepted Answer

$$0$$

Question 12

If $$s(t) is $$the distance from a starting point at time $$t$$, which formula gives the velocity $$v(t)$$?

Accepted Answer

$$v(t) = s'(t)$$

Question 13

For $$y = (3x^2$ + 1)^2$, what is y$$''$$?

Accepted Answer

$$78x^2 + 12$

Question 14

Which of the following is the correct Leibniz notation for the second derivative of $$y$$ with respect to $$x$$?

Accepted Answer

$$\frac{d^2$y}{dx^2$}$$

Question 15

If $$s(t) = 16t^2$, what is the velocity v(3.5$$)$$ after 3.5$ seconds?

Accepted Answer

$$112$$ 	ext{ feet/second}$

Question 16

If $$f(x) = x^5$ - 3x^4$ + x$$, what is $$f^{(4)}(x$$)$$?

Accepted Answer

$$120x - 72$$

Question 17

Which of the following statements is true about higher-order derivatives?

Accepted Answer

The $$nth $$derivative of a polynomial of degree $$m is $$zero for all $$n \u003e m.$$

Higher Order Derivatives: Concepts, Notation, and Applications

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