Question 1

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, find all critical points of $$f(x) on $$its entire domain.

Accepted Answer

$$x = 1, 3$$

Question 2

For the function $$f(x) = x^4$ - 4x^2$, at what x$-values does the graph have horizontal tangent lines?

Accepted Answer

$$x = 0, x = \pm \sqrt{2}$$

Question 3

The energy stored in a stretched spring is given by $$E = \frac{1}{2}kx^2$. If k = 2$ Joules/cm^2$ and the spring is being stretched at a rate of 3$ cm/sec, at what rate is the energy increasing when x = 10$ cm?

Accepted Answer

$$60$$ Joules/sec

Question 4

Find the equation of the tangent line to $$f(x) = x^2$ + 2x at $$the point $$(1, 3).$$

Accepted Answer

$$y = 4x - 1$$

Question 5

Given $$f(x) = \sin(x) + x^2$, find f$$''(x)$$.

Accepted Answer

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

Question 6

If $$f(x) = 3x^4$ - 8x^3$ + 6x^2$, find the absolute maximum and minimum values of f(x$$)$$ on the interval $$[-2, 0]$$.

Accepted Answer

Absolute MAX: $$f(0) = 0 at$$ $$x = 0$$; Absolute MIN: $$f(-2) = -80 at$$ $$x = -2$$

Question 7

If $$y = x^2$ e^x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 8

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

Accepted Answer

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

Question 9

If $$f(x) = x^3$ - 3x^2$ + 2x$$, what is the simplified form of $$f'(x)$$?

Accepted Answer

$$3x^2 - 6x + 2$

Question 10

Suppose $$y = \ln(x^2$ + 1). $$Find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 11

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

Accepted Answer

$$-\sin(x)$$

Question 12

Given $$y = x^2$ \sin(x)$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$2x \sin(x) + x^2$ \cos(x)$$

Question 13

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 14

Suppose $$y = 	an(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec^2$(x)$$

Question 15

If $$f(x) = x^2$ + 4x + 5$$, what is the equation of the tangent line at $$x = -1$$?

Accepted Answer

$$y = 2x + 8$$

Question 16

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

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 17

Given $$f(x) = x^3$ - 3x$$, find all $$x-$$values where $$f'(x) = 0.$$

Accepted Answer

$$x = -1, 1$$

Question 18

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

Accepted Answer

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

Question 19

Suppose $$f(x) = x^2$ e^x. $$What is $$f''(x)$$?

Accepted Answer

$$2e^x + 4x e^x + x^2$ e^x$$

Question 20

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

$$-e^{-0.6}$$

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