Q: Suppose the population of a bacteria culture at time $$t (in $$hours) is given by $$P(t) = 500e$$^{0.2t}$$. What is the instantaneous growth rate of the population at t = 5$ hours?

$$P'(5) = 100e$$^{0.2 \times 5}$$

Q: Find the derivative of $$f(x) = \dfrac{\tan(x)}{x}.$$

$$f'(x) = \dfrac{x \sec^2$(x) - \tan(x)}{x^2$}$$

Question 1

Let $$f(x) = 3x^4$ - 5x^2$ + 2x - 7. $$What is $$f''(x)$$, the second derivative of $$f(x)$$?

Accepted Answer

$$f''(x) = 12x^2$ - 10$$

Question 2

Suppose the population of a bacteria culture at time $$t (in $$hours) is given by $$P(t) = 500e$$^{0.2t}$$. What is the instantaneous growth rate of the population at t = 5$ hours?

Accepted Answer

$$P'(5) = 100e$$^{0.2 	imes 5}$$

Question 3

Given $$y = x^2$ \sin(x)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

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

Question 4

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V = 4t^2$ + 3t. At $$what time is the rate of water flow into the tank exactly $$35$$ liters per minute?

Accepted Answer

$$t = 4$$

Question 5

Find the derivative of $$f(x) = \dfrac{	an(x)}{x}.$$

Accepted Answer

$$f'(x) = \dfrac{x \sec^2$(x) - 	an(x)}{x^2$}$$

Question 6

If $$y^2$$ + $$x^2 = 25$, find $$\dfrac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

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

Question 7

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. How fast is the volume of the balloon increasing when the radius is $$5 cm$$? (Recall: $$V = \dfrac{4}{3}\pi r^3$)

Accepted Answer

$$200\pi cm$^3/$$min

Question 8

Find the equation of the tangent line to the curve $$y = x^3$ - 2x + 1 at $$the point where $$x = 2.$$

Accepted Answer

$$y = 10x - 15$$

Question 9

Let $$f(x) = \sin(x) \cos(x). $$What is $$f''(x)$$?

Accepted Answer

$$f''(x) = -4\sin(x) \cos(x)$$

Question 10

Given $$f(x) = (2x^2$ + 3x)^5$, use the chain rule to find f$$'(x)$$.

Accepted Answer

$$f'(x) = 5(2x^2$ + 3x)^4$ (4x + 3)$$

Question 11

A medication is administered such that its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. At what time is the concentration decreasing at a rate of 2$ mg/L per hour?

Accepted Answer

$$t = \ln(2.5)$$

Question 12

Find all critical points of $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 13

Let $$y = \sin(x^2$). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

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

Question 14

A ladder 10 ft long is leaning against a wall. If the bottom of the ladder slides away from the wall at 3 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom is 6 ft from the wall?

Accepted Answer

$$-2.25 ft/sec$$

Question 15

Find the absolute maximum value of $$f(x) = -x^2$ + 4x + 1 on $$the interval $$[0, 5].$$

Accepted Answer

$$f(2) = 5$$

Question 16

If $$y = 	an(3x)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$3 \sec^2$(3x)$$

Question 17

Given $$x^2 + xy$$ + $$y^2 = 7$, find $$\dfrac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\dfrac{dy}{dx} = -\dfrac{2x + y}{x + 2y}$$

Question 18

Which of the following points is a local minimum for $$f(x) = x^3$ - 6x^2$ + 9x + 1$$?

Accepted Answer

$$x = 3$$

Question 19

If $$f(x) = \dfrac{1}{x^2$ + 1}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -\dfrac{2x}{(x^2$ + 1)^2$}$$

Calculus Study Guide: Derivatives and Applications

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