$$e^x = \lim_{n \to \infty} e$$^{x_n}$$, where x_n$ is a sequence of rationals converging to 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 function $$f(x) = e^x$$?

Accepted Answer

$$f'(x) = e^x$$

Question 3

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

Accepted Answer

The function is multiplicative over addition

Question 4

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

Accepted Answer

$$\infty$$

Question 5

Which of the following is the correct formula for the derivative of $$\ln u(x)$$, where $$u(x) is a $$differentiable function?

Accepted Answer

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

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 NOT a property of the exponential function $$e^x$$?

Accepted Answer

$$e^x is $$always negative for $$x \u003c 0$$

Question 8

If $$x is $$irrational, how is $$e^x$$ defined?

Accepted Answer

$$e^x = \lim_{n 	o \infty} e$$^{x_n}$$, where x_n$ is a sequence of rationals converging to x$

Question 9

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

Accepted Answer

$$e^x = 1 + x + \frac{x^2$}{2!} + \frac{x^3$}{3!} + \frac{x^4$}{4!} + \cdots$$

Question 10

What is the value of $$\ln(1)$$?

Accepted Answer

$$0$$

Question 11

Which of the following is a correct property of the natural logarithm?

Accepted Answer

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

Question 12

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

Accepted Answer

Hyperbolic cosine

Question 13

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

Accepted Answer

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

Question 14

Which of the following is the inverse function of $$e^x$$?

Accepted Answer

$$\ln(x)$$

Question 15

For which value of $$x$$ does $$e^x$$ cross the $$y-$$axis?

Accepted Answer

$$x = 0$$

Question 16

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

Accepted Answer

It has a horizontal asymptote at $$y = 0$$

Question 17

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

Accepted Answer

$$2e^{2x}$$

Question 18

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

Accepted Answer

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

Question 19

If $$f(x) = \ln(e^x)$$, what is $$f(x)$$?

Accepted Answer

$$x$$

Question 20

Which of the following is a correct property of logarithms?

Accepted Answer

$$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$$

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

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