Question 1

Which of the following best describes the behavior of the function $$f(x) = a_0$ \cdot a^x$$ for $$a \u003e 1 as$$ $$x$$ increases?

Accepted Answer

The function increases without bound and is concave up.

Question 2

Given $$f(x) = a_0$ \cdot a^x$$ with $$0 \u003c a \u003c 1$$, what is the horizontal asymptote of the graph?

Accepted Answer

$$y = 0$$

Question 3

What is the derivative of $$f(x) = e$$^{3x^2}$$ with respect to x$?

Accepted Answer

$$6x e$$^{3x^2}$$

Question 4

If $$f(x) = 5e$$^{-2x}$$, what is f$$'(x)$$?

Accepted Answer

$$-10e^{-2x}$$

Question 5

A company purchases equipment for $$\17$$,500$$. Its value after t$ years is given by V(t) = 17$$,$$500e^{-0.0834t}$. What is the value of the equipment after 5 years? (Round to the nearest dollar.)

Accepted Answer

$$11,470$$

Question 6

For the function $$f(x) = e$$^{x^2 + 2x}$$, what is f$$'(x)$$?

Accepted Answer

$$(2x + 2)e$$^{x^2 + 2x}$$

Question 7

Which of the following statements is true about the graph of $$f(x) = a_0$ \cdot a^x$$ for $$a \u003e 1$$?

Accepted Answer

The graph is concave up and has a horizontal asymptote at $$y = 0.$$

Question 8

Differentiate $$y = x^2$ e$$^{3x}$$ with respect to x$.

Accepted Answer

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

Question 9

If $$f(x) = 2e$$^{5x}$$ - 3e$$^{-2x}$$, what is f$$'(x)$$?

Accepted Answer

$$10e^{5x}$$ + $$6e^{-2x}$$

Question 10

Given $$y = e$$^{x^2 - 4x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$(2x - 4)e$$^{x^2 - 4x}$$

Question 11

A medication's concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$-7.44$$

Question 12

Which of the following is the correct derivative of $$f(x) = e$$^{x}$$ + e$$^{-x}$$?

Accepted Answer

$$e^{x}$$ - $$e^{-x}$$

Question 13

If $$f(x) = 4x e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$4e^{2x} + 8x$$ $$e^{2x}$$

Question 14

Which of the following best describes the function $$f(x) = a_0$ \cdot a^x$$ for $$0 \u003c a \u003c 1 as$$ $$x$$ increases?

Accepted Answer

The function decreases and approaches zero.

Question 15

Differentiate $$y = 3e$$^{-4x}$$ + 2x$$ with respect to $$x.$$

Accepted Answer

$$-12e^{-4x} + 2$

Question 16

If $$f(x) = e$$^{x^3}$$, what is f$$'(x)$$?

Accepted Answer

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

Question 17

A population of bacteria grows according to $$P(t) = 200e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$16.49$$

Question 18

Which of the following is the correct derivative of $$f(x) = e$$^{2x - 5}$$?

Accepted Answer

$$2e^{2x - 5}$$

Question 19

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

Accepted Answer

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

Derivatives of Exponential (Base-e) Functions in Business Calculus

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