Inverse Functions and Their Derivatives

The video tutorial covers a range of calculus topics, starting with a warm-up for an upcoming quiz. It delves into logarithmic differentiation, various integration techniques, and a review of inverse functions. The tutorial also explains how to find derivatives of inverse functions and provides numerical examples for practice. The session is designed to prepare students for both the quiz and their unit portfolio, emphasizing key calculus concepts and problem-solving strategies.

What is the derivative of $3 \ln x$?

• 3x

• x/3

• 3/x

• ln(3x)

Tags

CCSS.HSF-BF.B.4C

In the context of inverse functions, what does the point (2, 1) on the original function become on the inverse function?

• (2, 2)

• (1, 2)

• (2, 3)

• (1, 1)

Tags

CCSS.HSF-BF.B.4A

If $f(x) = \sqrt{x^3 + 2x + 1}$, what is the first step to find $f^{-1}(x)$ at $x = 2$?

• Set $x = 2$

• Set $f(x) = 1$

• Set $f^{-1}(x) = 2$

• Set $f(x) = 2$

Tags

CCSS.HSF-BF.B.4A

Given $f(x) = \frac{1}{x-1}$, what is the derivative of the inverse function at $x = 2$?

• 4

• -1/4

• 1/4

• -4

Tags

CCSS.HSF-BF.B.4C

What is the slope of the inverse function at the point (3, 2) if the slope of the original function at (2, 3) is 4?

• 1/4

• 4

• -4

• -1/4

Tags

CCSS.HSF.IF.A.2

If $f(x) = x^3 + x - 1$, what is the value of $x$ when $f(x) = 3$?

• 4

• 3

• 1

• 2

Tags

CCSS.HSF-BF.B.4C

Given $f(x) = x^3 + x - 1$, what is the derivative of $f$ at $x = 2$?

• 3

• 1

• 4

• 2

Tags

CCSS.HSF-BF.B.4C

If $f(x) = x^3 + x - 1$, what is the slope of the inverse function at $x = 3$?

• 1/4

• 4

• -4

• -1/4

Tags

CCSS.HSF-BF.B.4C

What is the slope of $f$ at $x = 3$ if $f(3) = 2$ and $f'(3) = -1/2$?

• 2

• -2

• 1/2

• -1/2

Tags

CCSS.HSF-BF.B.4C

What is the slope of the inverse function at $x = 2$ if the slope of the original function at $x = 3$ is -1/2?

• 2

• 1/2

• -2

• -1/2

Tags

CCSS.HSF-BF.B.4C