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

$$5 \times 100e$$^{0.05 \times 10}$$

Question 1

Given the quadratic function $$f(x) = 2x^2$ - 3x + 1$$, what is the value of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$4x + 2h - 3$$

Question 2

Use the limit definition of the derivative to find $$f'(x)$$ for $$f(x) = x^2$. Which of the following is correct?

Accepted Answer

$$2x$$

Question 3

The position $$s(t) of a $$car in meters is given by $$s(t) = 5t^2$ + 2t. $$What does the derivative $$s'(t)$$ represent in this context?

Accepted Answer

The instantaneous velocity of the car at time $$t$$

Question 4

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

Accepted Answer

$$5 	imes 100e$$^{0.05 	imes 10}$$

Question 5

Given $$f(x) = 3x^4$ - 2x^2$ + 7$$, what is $$f''(x)$$, the second derivative?

Accepted Answer

$$12x^2 - 4$

Question 6

If $$f(x) = \ln(3x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{6x}{3x^2$ + 1}$$

Question 7

A medication's concentration in the bloodstream is modeled by $$C(t) = 8e$$^{-0.2t}$$. What is the rate at which the concentration is changing at t = 5$ hours?

Accepted Answer

$$-1.6e^{-1}$$

Question 8

Find the equation of the tangent line to $$f(x) = x^3$ - 2x at$$ $$x = 1.$$

Accepted Answer

Question 9

Let $$f(x) = x^2$ and g(x) = e^x$. Using the product rule, what is $$\frac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

$$x^2e^x + 2xe^x$

Question 10

Given $$h(x) = \frac{2x^3$ + 1}{x^2$}$$, what is $$h'(x)$$ using the quotient rule?

Accepted Answer

$$\frac{6x^2$x^2$$ - $$(2x^3 + 1)2x$$}{$$x^4$$}$

Question 11

If $$f(x) = 5x^2$ + 3x$$ and $$g(x) = 2x - 1$$, what is $$\frac{d}{dx}[f(x) + g(x)]$$?

Accepted Answer

$$10x + 3 + 2$$

Question 12

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

Accepted Answer

$$2e^{2x}$$

Question 13

Given $$F(x) = 7$$, what is $$F'(x)$$?

Accepted Answer

$$0$$

Question 14

Suppose $$f(x) = (3x^2$ + 2x)^5$. Using the general power rule, what is f$$'(x)$$?

Accepted Answer

$$5(3x^2$$ + $$2x)^4$$ \cdot (6x + 2)$$

Question 15

A function $$f(x, y) = x^2$y + 3y^2$. What is the partial derivative $$\frac{\partial f}{\partial x}$$?

Accepted Answer

$$2xy$$

Question 16

Given $$f(x, y) = x^2$y + 3y^2$, what is $$\frac{\partial f}{\partial y}$$ evaluated at (1$$, 2)$$?

Accepted Answer

$$13$$

Question 17

A chemical reaction's rate is modeled by $$R(t) = 4t^3$ - 2t + 5. $$What is the instantaneous rate of change of the reaction at $$t = 2$$?

Accepted Answer

$$92$$

Question 18

If $$f(x) = 2x^3$ - 5x^2$ + 4x - 7$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$-3$$

Question 19

Let $$f(x) = \ln(x^2$ + 1). $$What is $$f''(x)$$, the second derivative?

Accepted Answer

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

Question 20

Suppose the temperature $$T at a $$point $$(x, y) on a $$metal plate is given by $$T(x, y) = 2x^2$y + 3y^3$. What is the rate of change of temperature with respect to y$ at the point (1$$, 1)$$?

Accepted Answer

$$8$$

Calculus I: Derivatives and Their Applications

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