Q: A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous growth rate at t = 10$ hours?

$$P'(10) = 5e$$^{0.5}$$ \times 100 \approx 824$$

Question 1

Suppose the position of a car along a straight road is given by $$s(t) = 3t^3$ - 5t^2$ + 2t$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$18 m/s$$

Question 2

Let $$f(x) = x^4$ - 4x^3$ + 6x^2$. Find all x$-values where the tangent line to the graph of f$ has a slope of 0$.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 3

Find the second derivative of $$f(x) = \sin(x) \cos(x).$$

Accepted Answer

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

Question 4

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous growth rate at t = 10$ hours?

Accepted Answer

$$P'(10) = 5e$$^{0.5}$$ 	imes 100 \approx 824$$

Question 5

Given $$f(x) = x^2$ \sin(x)$$, find $$f'(x)$$ using the product rule.

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Given $$x^2$$ + $$y^2 = 25$, find $$\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 of the balloon increasing when the radius is $$5 cm$$? (Recall: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Which of the following statements about absolute and local extrema is TRUE?

Accepted Answer

An absolute minimum is always a local minimum.

Question 12

Given $$y^3 + xy = 7$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{y}{3y^2$ + x}$$

Question 13

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

Accepted Answer

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

Question 14

A tank is being filled with water at a rate of $$10$$ liters per minute. The volume $$V of $$water in the tank at time $$t is$$ $$V = 10t. $$What is the instantaneous rate of change of the water level at $$t = 5$$ minutes?

Accepted Answer

$$10$$ liters/min

Question 15

Find the derivative of $$f(x) = \frac{\sin(x)}{x}$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{x\cos(x) - \sin(x)}{x^2$}$$

Question 16

Suppose $$y = \cos(3x). $$What is the second derivative $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

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

Question 17

A ladder $$10 ft $$long is leaning against a wall. If the bottom of the ladder slides away from the wall at $$1 ft/sec$$, how fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-0.8 ft/sec$$

Question 18

Find the steady-state population for the logistic growth model $$P(t) = \frac{500}{1 + 9e$$^{-0.2t}$$}.$$

Accepted Answer

$$500$$

Question 19

Given $$f(x) = x^2$ e$$^{x}$$, find all critical points.

Accepted Answer

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

Calculus Study Guide: Derivatives and Applications

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