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Calculus AB : Calculus AB

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45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Use Derivatives Of Trig Functions

True or False: When the derivative of tan(x) is negative but increasing, then the function tan(x) is increasing.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Even though we are working with trigonometric functions, the rules of derivatives remain the same. Since the graph of a derivative function is the graph of the rate of change of the original function, then when the graph of the derivative function is negative, the original function is decreasing. Even when the derivative is increasing, if it is negative the original function is still decreasing, just by less of a factor. When the derivative function is at zero, then the original function is constant. When the derivative function is positive, then the original function is increasing.

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Example Question #2 : Use Derivatives Of Trig Functions

Find the derivative of f(x) = cot(5x^2) .

Possible Answers:

f'(x) = 5x^2csc^2(5x^2)

f'(x) = 5csc^2(5x^2)

f'(x) = csc^2(5x^2)

f'(x) = 10xcsc^2(5x^2)

Correct answer:

f'(x) = 10xcsc^2(5x^2)

Explanation:

We need to use the Chain Rule to take both the derivative of the trigonometric function and the quantity within the trig function.

f(x) = cot(5x^2)

$f'(x) = csc^2(5x^2) * \frac{d}{dx}[5x^2]$

f'(x) = 10xcsc^2(5x^2)

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Example Question #61 : Basic Differentiation Rules

What is the derivative of f(x) = tan^2(x) ?

Possible Answers:

f'(x) = 2tan(x)

f'(x) = sec^2(x)tan(x)

f'(x) = 2sec^2(x)

f'(x) = 2sec^2(x)tan(x)

Correct answer:

f'(x) = 2sec^2(x)tan(x)

Explanation:

Recall that the derivative of the tangent function is sec(x) . So we need to take the derivative of the exponent of tan^2(x) and we also need the derivative of the actual tangent function.

f(x) = tan^2(x)

$f'(x) = 2tan(x) *\frac{d}{dx}[tan(x)]$

f'(x) = 2sec^2(x)tan(x)

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Example Question #1 : Differentiate Inverse Functions

Which expression correctly identifies the inverse of f (x) = 2x-4 ?

Possible Answers:

$f^{-1}(x) =\frac{1}{2}x+4$

$f^{-1}(x) =\frac{1}{2}x-2$

$f^{-1}(x) =x+2$

$f^{-1}(x) =\frac{1}{2}x+2$

Correct answer:

$f^{-1}(x) =\frac{1}{2}x+2$

Explanation:

The inverse of a function can be found by substituting yvariables for the variables found in the function, then setting the function equal to . By next isolating , the inverse function is written. Then, the notation $f^{-1}$ is used to describe the newly written function as being the inverse of the original function. The answer choice “ $f^{-1}(x) =\frac{1}{2}x+2$ ” is correct.

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Example Question #1 : Differentiate Inverse Functions

Which of the following correctly identifies the derivative of an inverse function?

Possible Answers:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$

$(f^{-1})'(y) = \frac{1}{f'(f^{-1}(x))}$

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(y))}$

$(f^{-1})'(x) = \frac{2}{f'(f^{-1}(x))}$

Correct answer:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$

Explanation:

This question asks you to recognize the correct notation of a differentiating inverse functions problem. First, it is key to recognize that the equation needs to have the same variable throughout, thus eliminating the answer choices $(f^{-1})'(y) = \frac{1}{f'(f^{-1}(x)) }$ and $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(y))}$ . Next, there should be no constants in the correct equation; thus, $(f^{-1})'(x) = \frac{2}{f'(f^{-1}(x))}$ is incorrect. The correct choice is $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

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Example Question #1 : Differentiate Inverse Functions

Find $(f^{-1})'(1)$ given f (x) = x^5+2x+1 .

Possible Answers:

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{2}{3}$

Correct answer:

$\frac{1}{2}$

Explanation:

Let f (x) = x^5+2x+1

It is important to recognize the relationship between a function and its inverse to solve.

If f (0) = (0)^5+2(0)+1=1 , solving for the inverse function will produce $f^{-1}(1)=0$ .

To find the derivative of an inverse function, use: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$

$(f^{-1})'(1) = \frac{1}{f'(f^{-1}(1))}=\frac{1}{f' (0)}=\frac{1}{5(0)^4+ 2}=\frac{1}{0 + 2}=\frac{1}{2}$

Therefore, $(f^{-1})'(1)=\frac{1}{2}$

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Example Question #2 : Differentiate Inverse Functions

Let f (x) =x^6 .Find $(f^{-1})'(x)$ .

Possible Answers:

$\frac{6}{x}$

$\frac{1}{6x}$

$\frac{1}{6x^5_6}$

$\frac{1}{4x^5_6}$

Correct answer:

$\frac{1}{6x^5_6}$

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x) =x^6 \rightarrow f^{-1} (x) =\sqrt[6]{x}$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\sqrt[6]{x})}$

After substituting in $\sqrt[6]{x}$ for $f^{-1} (x)$ , the derivative of f (x), or f'(x) =6x^5 , is applied:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\sqrt[6]{x})}=\frac{1}{6(\sqrt[6]{x})^5}=\frac{1}{6x^5_6}$

Therefore, the correct answer is $(f^{-1})'(x)=\frac{1}{6x^5_6}$

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Example Question #1 : Differentiate Inverse Functions

Let f (x) =e^x . Find $(f^{-1})'(x)$ .

Possible Answers:

ln(x)

$\frac{1}{x}$

$xe^{x-1}$

e^x

Correct answer:

$\frac{1}{x}$

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x) =e^x \rightarrow f^{-1} (x) =ln(x)$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(ln(x))}$

After substituting in $ln(x) for f^{-1} (x)$ , the derivative of f (x) , or f'(x) =e^x , is applied:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(ln(x))}=\frac{1}{e^{ln(x)}}=\frac{1}{x}$

Therefore, the correct answer is $(f^{-1})'(x)=\frac{1}{x}$

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Example Question #2 : Differentiate Inverse Functions

Let $f (x) =\sqrt[3]{x}$ . Find $(f^{-1})'(x)$ .

Possible Answers:

2x^3

$\frac{3}{x^2}$

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

3x^2

Correct answer:

3x^2

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x)=\sqrt[3]{x} \rightarrow f^{-1} (x)=x^3$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(x^3)}$

After substituting in x^3 for $f^{-1} (x)$ , the derivative of f (x) , or $f'(x)=(\frac{1}{3})(x^{-\frac{2}{3}})$ , is applied:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{(\frac{1}{3})(x^3)^{-\frac{2}{3}}}=\frac{3}{x^{-2}}=3x^2$

Therefore, the correct answer is $(f^{-1})'(x)=3x^2$ .

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Example Question #2 : Differentiate Inverse Functions

Let f (x) =2-2x . Find $(f^{-1})'(x)$ .

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{2}$

Correct answer:

$-\frac{1}{2}$

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x)=2-2x \rightarrow f^{-1} (x)=1-\frac{1}{2}x$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(1-\frac{1}{2}x)}$

After substituting in $1-\frac{1}{2}x$ for $f^{-1} (x)$ , the derivative of f (x) , or f'(x)= -2 , is applied:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}= -\frac{1}{2}$

Therefore, the correct answer is $(f^{-1})'(x)=-\frac{1}{2}$ .

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Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Use Derivatives Of Trig Functions

Example Question #2 : Use Derivatives Of Trig Functions

Example Question #61 : Basic Differentiation Rules

Example Question #1 : Differentiate Inverse Functions

Example Question #1 : Differentiate Inverse Functions

Example Question #1 : Differentiate Inverse Functions

Example Question #2 : Differentiate Inverse Functions

Example Question #1 : Differentiate Inverse Functions

Example Question #2 : Differentiate Inverse Functions

Example Question #2 : Differentiate Inverse Functions

All Calculus AB Resources

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