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$?

Accepted Answer

$$P'(5) = 100e$$

Question 3

Find the derivative of $$y = x^2$ \sin(x)$$ using the product rule.

Accepted Answer

$$y' = 2x\sin(x) + x^2$\cos(x)$$

Question 4

Given $$f(x) = \frac{2x^3$ - 5x}{x^2$}$$, find $$f'(x)$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{6x^2$x^2$$ - $$(2x^3 - 5x)2x$$}{$$x^4$$}$

Question 5

Find the derivative of $$y = 	an(x).$$

Accepted Answer

$$y' = \sec^2$(x)$$

Question 6

If $$y = \sin(3x)$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\frac{d^2$y}{dx^2$} = -9\sin(3x)$$

Question 7

Let $$y = (2x^2$ + 1)^5$. Find $$\frac{dy}{dx}$$ using the chain rule.

Accepted Answer

$$\frac{dy}{dx} = 20x(2x^2$ + 1)^4$

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$\frac{dV}{dt} = 200\pi cm$^3/$$min

Question 10

Find the $$x-$$values where the tangent line to $$f(x) = x^3$ - 3x^2$ + 2$$ has slope $$0.$$

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 11

Given $$f(x) = x^4$ - 4x^2$, find all critical points.

Accepted Answer

$$x = 0, x = 2, x = -2$$

Question 12

A tank is being filled with water at a rate of $$5 L/$$min. The height $$h of $$the water is related to the volume $$V by$$ $$V = \pi r^2$ h$$, where $$r = 2 m. $$How fast is the height rising when $$h = 3 m$$?

Accepted Answer

$$\frac{dh}{dt} = \frac{5}{4\pi} m/$$min

Question 13

Find the equation of the tangent line to the curve $$y = x^2$ e^x at $$the point where $$x = 0.$$

Accepted Answer

$$y = x$$

Question 14

Which of the following $$x-$$values is a location of a local maximum for $$f(x) = -x^3$ + 3x^2$ + 1$$?

Accepted Answer

$$x = 2$$

Question 15

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

Accepted Answer

$$\frac{dy}{dx} = -2x\sin(x^2$)$$

Question 16

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

Accepted Answer

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

Question 17

Find the steady-state population for the logistic growth model $$P(t) = \frac{1000}{1 + 9e$$^{-0.5t}$$}.$$

Accepted Answer

$$1000$$

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

$$-2 ft/sec$$

Calculus Study Guide: Derivatives and Applications

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