Q: A chemical reaction's temperature (in $$^\circC) is $$given by $$T(t) = 100 - 80e$$^{-0.2t}$$, where t$ is in minutes. What is the rate of change of temperature at t = 5$ minutes?

$$16e^{-1}$$ \text{ $$^\circC/$$min}$

Q: If $$y = e$$^{\tan(x)}$$, what is $$\frac{dy}{dx}$$?

$$e^{\tan(x)}$$ \$$sec^2(x$$)$$

Question 1

A ball is thrown straight up from the ground with an initial velocity of $$20 m/s. $$Using $$g = 9.8 m/s$^2$$, what is the velocity of the ball after $$2$$ seconds? (Assume upward is positive.)

Accepted Answer

$$0.4 m/s$

Question 2

Let $$f(x) = x^3$ e$$^{2x}$$. What is f$$'(x)$$?

Accepted Answer

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

Question 3

The population $$P of a $$bacteria culture at time $$t (in $$hours) is given by $$P(t) = 500e$$^{0.3t}$$. What is the rate of change of the population at t = 4$ hours?

Accepted Answer

$$150e^{1.2}$$

Question 4

Find the equation of the tangent line to the curve $$y = \sin(x) + x^2$ at x = 0$.

Accepted Answer

$$y = x$$

Question 5

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

Accepted Answer

$$6x \sec^2$(3x^2$$)$

Question 6

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$ mg/L, where t$ is in hours. What is the rate at which the concentration is changing at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$ 	ext{ mg/L/hr}$

Question 7

Let $$f(x) = \frac{2x^2$ + 3x}{x^2$ - 1}. $$What is $$f'(x)$$?

Accepted Answer

$$\frac{(4x + 3)(x^2$ - 1) - (2x^2$ + 3x)(2x)}{(x^2$ - 1)^2$}$$

Question 8

Suppose the position of a car is given by $$s(t) = 4t^3$ - 2t^2$ + t (in $$meters, $$t in $$seconds). What is the acceleration of the car at $$t = 2$$ seconds?

Accepted Answer

$$44$$ 	ext{ m/s}^2$

Question 9

If $$f(x) = \sin(x) \cdot e^x$$, what is $$f'(x)$$?

Accepted Answer

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

Question 10

A chemical reaction's temperature (in $$^\circC) is $$given by $$T(t) = 100 - 80e$$^{-0.2t}$$, where t$ is in minutes. What is the rate of change of temperature at t = 5$ minutes?

Accepted Answer

$$16e^{-1}$$ 	ext{ $$^\circC/$$min}$

Question 11

Let $$y = (2x^2$ + 1)^5$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$20x(2x^2$$ + $$1)^4$$

Question 12

A company's revenue $$R (in $$dollars) from selling $$x$$ units is $$R(x) = 100x - 0.5x^2$. What is the marginal revenue when x = 20$?

Accepted Answer

$$80$$

Question 13

If $$f(x) = \cos(x^2$)$$, what is $$f'(x)$$?

Accepted Answer

$$-2x \sin(x^2$)$$

Question 14

The height $$h (in $$meters) of a projectile at time $$t (in $$seconds) is $$h(t) = -4.9t^2$ + 30t + 2. $$What is the instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$0.6$$ 	ext{ m/s}$

Question 15

Let $$f(x) = \frac{\sin(x)}{x}. $$What is $$f'(x)$$ for $$x 
eq 0$$?

Accepted Answer

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

Question 16

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

Accepted Answer

$$e^{	an(x)}$$ \$$sec^2(x$$)$$

Question 17

A bacteria population grows according to $$P(t) = 2000e$$^{0.1t}$$, where t$ is in hours. How fast is the population growing at t = 10$ hours?

Accepted Answer

$$200e^{1}$$

Question 18

Let $$f(x) = x^2$ \cos(x). $$What is $$f'(x)$$?

Accepted Answer

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

Question 19

A tank is being filled with water so that the volume at time $$t (in $$minutes) is $$V(t) = 5t^2$ + 3t. $$What is the rate at which water is being added at $$t = 4$$ minutes?

Accepted Answer

$$43$$ 	ext{ L/min}$

Calculus Study Guide: Techniques of Differentiation and Applications

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