Question 1

Let $$f(x) = x^3$ - 6x^2$ + 9x + 1. On $$which interval is $$f(x)$$ decreasing?

Accepted Answer

$$(1, 3)$$

Question 2

Suppose $$f(x) = e$$^{2x}$$ - 4e^x. $$Find all critical points and use the First Derivative Test to classify them.

Accepted Answer

Critical point at $$x = \ln 2$$, local maximum

Question 3

Given $$f(x) = x^4$ - 4x^2$, use the Second Derivative Test to determine the nature of the critical point at x = 0$.

Accepted Answer

Inconclusive

Question 4

A chemical reaction's temperature (in $$^\circC) at $$time $$t (in $$minutes) is modeled by $$T(t) = -t^2$ + 4t + 10$$ for $$t in$$ $$[0, 5]. $$What is the absolute maximum temperature?

Accepted Answer

$$T(2) = 14$$

Question 5

Use differentials to approximate $$\sqrt{25.2}.$$

Accepted Answer

$$5.04$$

Question 6

A radioactive substance has a half-life of 10 years. What is the value of $$k in $$the exponential decay model $$N(t) = N_0$ e$$^{kt}$$?

Accepted Answer

$$k = -\dfrac{\ln 2}{10}$$

Question 7

Find $$\dfrac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$.

Accepted Answer

$$\dfrac{dy}{dx} = -\dfrac{x}{y}$$

Question 8

Differentiate $$f(x) = x^2$ e$$^{3x}$$.

Accepted Answer

$$2x e$$^{3x}$$ + 3x^2$ e$$^{3x}$$

Question 9

Expand $$\ln(2x^3$)$$ using logarithmic properties.

Accepted Answer

$$\ln 2 + 3\ln x$$

Question 10

For $$f(x) = x^4$ - 8x^2$, on which interval is f(x$$)$$ concave up?

Accepted Answer

$$(-\infty, -2) \cup (2, \infty)$$

Question 11

Find all inflection points of $$f(x) = x^3$ - 6x^2$ + 9x + 1.$$

Accepted Answer

$$(2, 3)$$

Question 12

A company wants to build a rectangular enclosure next to a river (no fence needed along the river) using 120 meters of fencing. What is the maximum area that can be enclosed?

Accepted Answer

$$1800 m$^2$$

Question 13

Given $$f(x) = \ln(x^2$ + 1)$$, find $$f'(x).$$

Accepted Answer

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

Question 14

Suppose $$f(x) = \dfrac{x}{x^2$ + 1}. $$Find all critical points.

Accepted Answer

$$x = 0$$

Question 15

If $$y = x^x$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$x^x (1 + \ln x)$$

Question 16

Combine $$2\ln x - \ln y$$ into a single logarithm.

Accepted Answer

$$\ln \left(\dfrac{x^2$}{y}\right)$$

Question 17

A population of bacteria doubles every 3 hours. If the initial population is 100, what is the population after 9 hours?

Accepted Answer

$$800$$

Question 18

Find the equation of the tangent line to $$y^2$$ + $$x^2 = 10$ at the point (1$$, 3)$$.

Accepted Answer

$$y - 3 = -\dfrac{1}{3}(x - 1)$$

Question 19

Suppose $$f(x) = \dfrac{e^x}{x}. $$Find $$f'(x).$$

Accepted Answer

$$\dfrac{e^x(x - 1)}{x^2$}$$

Critical Points, Derivative Tests, and Applications in Calculus

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