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AP Calculus AB: 6.3.1 The Exponential and Natural Log Functions

Mathematics12 CardsCreated 3 months ago

This flashcard set introduces the exponential function 𝑒 π‘₯ e x and the natural logarithm function ln π‘₯l nx, highlighting their properties, graphs, and inverse relationship. It also touches on their domains, asymptotes, and basic derivatives, offering conceptual and visual understanding of these foundational transcendental functions.

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The Exponential and Natural Log Functions

  • Raising the number e to the power x produces the exponential function. It is a positive, increasing function.

  • Taking the log to the base e of x (log e x) produces the natural log function, noted ln x. It is an increasing function defined only for positive x-values.

  • The exponential and natural log functions are inverses of each other.

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Key Terms

Term
Definition

The Exponential and Natural Log Functions

  • Raising the number e to the power x produces the exponential function. It is a positive, increasing function.

  • Taking the log...

note

  • On the left, the graph of the exponential function (e x ) shows that it is positive and increasing. The x-axis is a left horizontal asympto...

For which of these values is the natural log function not defined?

cosΟ€

Which of the following is the graph of y =2e^x+1?

This exponential graph is increasing and passes through the point (0, 3).

Which is the largest among the following expressions?

ln 10

If y(x)=e^ln(sinx+2), what is dy/dx?

cos x

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AP Calculus AB: 6.3.1 The Exponential and Natural Log Functions
TermDefinition

The Exponential and Natural Log Functions

  • Raising the number e to the power x produces the exponential function. It is a positive, increasing function.

  • Taking the log to the base e of x (log e x) produces the natural log function, noted ln x. It is an increasing function defined only for positive x-values.

  • The exponential and natural log functions are inverses of each other.

note

  • On the left, the graph of the exponential function (e x ) shows that it is positive and increasing. The x-axis is a left horizontal asymptote for the curve at.

  • On the right, the graph of the natural log function (ln x)
    shows that it is increasing. It is only defined for positive
    x-values, in contrast to the exponential function. The y-axis is a vertical asymptote for the curve at.

  • The exponential and natural log functions are inverses of each other. When you compose them in either order, they cancel each other out.

  • Some mathematicians define ln x as the definite integral from one to x of the function 1/t.

  • The derivative of e^x is e^x .

  • The derivative of ln x is 1/x.

For which of these values is the natural log function not defined?

cosΟ€

Which of the following is the graph of y =2e^x+1?

This exponential graph is increasing and passes through the point (0, 3).

Which is the largest among the following expressions?

ln 10

If y(x)=e^ln(sinx+2), what is dy/dx?

cos x

Evaluate:

d/dx(e^2x+ln(2x))

2e^2x+1/x

Which of the following is the graph of

y = lnβ€Š(2x) + 1, for x > 0?

This logarithmic graph is increasing and passes through the point (0.5, 1).

How many of the following expressions have the same value as 2? e^ln2, ln(e^2), ln2, 1 + ln e

3

Which is the smallest among the following expressions?

ln 3

Given the following expressions: e2, ln(1/2), ln(e), and sin 0, which of the following is the order of their values from the smallest to the largest?

ln(1/2), sin0, ln(e), e

Which of the following is equivalent to d/dx(e^cosx)?

βˆ’e^cosx sinx