$$\lim_{h \to 0} \frac{e$$^{x+h}$$ - e^x}{h} = e^x$$

Question 1

Which of the following is the correct definition of the exponential function $$a^x$$ for $$a \u003e 0$$ and any real $$x$$?

Accepted Answer

$$a^x = e$$^{x \ln a}$$

Question 2

What is the derivative of the natural logarithm function $$\ln x$$ with respect to $$x$$?

Accepted Answer

$$\frac{d}{dx} \ln x = \frac{1}{x}$$

Question 3

If $$x is a $$rational number, $$x = \frac{p}{q}$$ with $$q \u003e 0$$, which of the following correctly expresses $$e^x$$?

Accepted Answer

$$e^x = e$$^{p/q}$$ = \sqrt[q]{e^p}$$

Question 4

Which property of the exponential function $$e^x is $$illustrated by the equation $$e^{x+y} = e^x e^y$?

Accepted Answer

The exponential function is multiplicative over addition.

Question 5

Evaluate the limit $$\lim_{x 	o \infty} e^x.$$

Accepted Answer

$$\infty$$

Question 6

Given $$u(x) = x^2$ + 2x + 1$$, what is $$\frac{d}{dx} \ln u(x)$$?

Accepted Answer

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

Question 7

Which of the following is the correct series expansion for $$e^x$$?

Accepted Answer

$$e^x = 1 + x + \frac{x^2$}{2!} + \frac{x^3$}{3!} + \cdots$$

Question 8

The function $$f(x) = \frac{e^x + e$$^{-x}$$}{2} is $$known as:

Accepted Answer

The hyperbolic cosine

Question 9

Which of the following is the correct derivative of $$e^x$$?

Accepted Answer

$$\frac{d}{dx} e^x = e^x$$

Question 10

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

Accepted Answer

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

Question 11

Which of the following is NOT a property of the natural logarithm $$\ln(x)$$?

Accepted Answer

$$\ln(a + b) = \ln a + \ln b$$

Question 12

Suppose the amount of a radioactive substance at time $$t is$$ $$N(t) = N_0$ e$$^{-kt}$$. What is the half-life T$ in terms of k$?

Accepted Answer

$$T = \frac{\ln 2}{k}$$

Question 13

Which of the following statements about the graph of $$y = e^x is $$true?

Accepted Answer

It passes through the point $$(0, 1).$$

Question 14

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

Accepted Answer

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

Question 15

Which of the following is the correct application of the chain rule to $$\frac{d}{dx} \ln(u(x))$$?

Accepted Answer

$$\frac{d}{dx} \ln(u(x)) = \frac{u'(x)}{u(x)}$$

Question 16

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

Accepted Answer

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

Question 17

Which of the following is the correct limit definition of the derivative of $$e^x$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{e$$^{x+h}$$ - e^x}{h} = e^x$$

Question 18

If $$f(x) = \ln(x)$$, which of the following is true about its domain?

Accepted Answer

The domain is $$x \u003e 0.$$

Question 19

Which of the following is a correct property of $$e^x$$?

Accepted Answer

$$e^{-x}$$ = \frac{1}{e^x}$$

Euler’s Exponential Function and the Natural Logarithm: Definitions, Properties, and Calculus Applications

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