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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #5 : Understanding The Limiting Process.

Differentiate: f(x)=(5x-3)(2x+1)

Possible Answers:

10x^2-x-3

20x-1

20x+1

Correct answer:

20x-1

Explanation:

Use the product rule to find the derivative of the function.

f(x)=(5x-3)(2x+1)

f'(x)= (5x-3)*2+(2x+1)*5

f'(x)= 10x-6+10x+5=20x-1

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Example Question #6 : Understanding The Limiting Process.

Differentiate: $y=e^{x^2}$

Possible Answers:

$2xe^{x^2}$

$2x^3e^{x^3}$

2e^x

$2e^{x^2}$

Correct answer:

$2xe^{x^2}$

Explanation:

The derivative of any function of e to any exponent is equal to the function multiplied by the derivative of the exponent.

$f'(x)=e^{x^2}*2x$

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Example Question #7 : Understanding The Limiting Process.

Find the second derivative of $y=ln(\frac{x^2}{x^2+1})$ .

Possible Answers:

$\frac{-2}{(x^3+x)^2}$

$\frac{2x}{x^4+x^2}$

$\frac{-2(3x^2+1)}{(x^3+x)^2}$

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

Correct answer:

$\frac{-2(3x^2+1)}{(x^3+x)^2}$

Explanation:

y=ln(x^2)-ln(x^2+1)

$y'= \frac{1}{x^2}*2x-\frac{1}{x^2+1}*2x$

$\frac{2x}{x^2}-\frac{2x}{x^2+1}= \frac{2x^3+2x-2x^3}{x^2*(x^2+1)}= \frac{2x}{x^4+x^2}$

Factoring out an x gives you $\frac{2}{x^3+x}$ .

$y''=2(x^3+x)^{-1}=-2(x^3+x)^{-2}*(3x^2+1)=\frac{-2(3x^2+1)}{(x^3+x)^2}$

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Example Question #181 : Functions, Graphs, And Limits

What is the derivative of $(2+3cos(3x))^\pi$ ?

Possible Answers:

$3\pi(2+cos(3x))^{\pi-1}sin(3x)$

$-3\pi(2+cos(3x))^{\pi-1}$

$-3\pi(2+cos(3x))^{\pi-1}sin(3x)$

$-3\pi(2+cos(3x))^{\pi-1}cos(3x)$

$3\pi(2+cos(3x))^{\pi-1}cos(3x)$

Correct answer:

$-3\pi(2+cos(3x))^{\pi-1}sin(3x)$

Explanation:

Need to use the power rule which states: $\frac{d}{dx}u^n=nu^{n-1}\frac{du}{dx}$

In our problem $\frac{du}{dx}=-3sin(3x)$

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Example Question #11 : Understanding The Limiting Process.

Consider:

$f(x) = x^{99} - x^{66}$

The 99th derivative of f(x) is:

Possible Answers:

99! + 66!

99!

99! - 66!

99!(x)

Correct answer:

99!

Explanation:

For $f(x) = x^{n}$ , the nth derivative is . As an example, consider $f(x) = x^{3}$ . The first derivative is $3x^{2}$ , the second derivative is $3*2x\: (or\: 6x)$ , and the third derivative is $3*2*1,\: or \: 3!$ . For the question being asked, the 99th derivative of $x^{99}$ would be 99! . The 66th derivative of $x^{66}$ would be 66! , and any higher derivative would be zero, since the derivative of any constant is zero. Thus, for the given function, the 99th derivative is 99! .

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Example Question #11 : Understanding The Limiting Process.

Consider the function $f(x) = 4x^{2} - sin(x) - 10x$ .

Which of the following is true when $x = \pi$ ?

Possible Answers:

f(x) < 0 and is increasing and concave up.

f(x) < 0 and is increasing and concave down.

f(x) > 0 and is increasing and concave up.

f(x) > 0 and is increasing and concave down.

f(x) > 0 and is decreasing and concave up.

Correct answer:

f(x) > 0 and is increasing and concave up.

Explanation:

$f(\pi ) = 4\pi^{2} - sin(\pi ) - 10\pi > 0$

$f'(\pi ) = 8\pi - cos(\pi) -10 > 0$ , meaning f(x) is increasing when x = $\pi$ .

$f''(\pi ) = 8 + sin(\pi ) > 0$ , meaning f(x) is concave up when x = $\pi$ .

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Example Question #11 : Understanding The Limiting Process.

Find the derivative of y=(3x^2+9)^2 .

Possible Answers:

$f{}'(x)=2(3x^2-9)$

$f{}'(x)=12(3x^2+9)$

f'(x)=12x(3x^2+9)

$f{}'(x)=6x^2$

Correct answer:

f'(x)=12x(3x^2+9)

Explanation:

To find the derivative of this expression, you must use the chain rule. This means you take the exponent of the binomial and multiply it by the coefficient in front of the binomial (1, in this case). Then, decrease the exponent of the binomial by 1. Lastly, find the derivative of the binomial.

Thus, your answer is:

2(3x^2+9)(6x) or 12x(3x^2+9) .

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Example Question #51 : Limits Of Functions (Including One Sided Limits)

Find the derivative of:

$y=\sqrt[3]{(3x^3+4)^2}$

Possible Answers:

$f{}'(x)=6x(\sqrt[3]{3x^3+4})$

$f{}'(x)=\frac{2}{3}(3x^3+4)$

$f{}'(x)=\frac{6x^2}{\sqrt[3]{3x^3+4}}$

$f{}'(x)=(3x^3+4)^{\frac{2}{3}}$

Correct answer:

$f{}'(x)=\frac{6x^2}{\sqrt[3]{3x^3+4}}$

Explanation:

This problem involves the chain rule for derivatives. However, you must first rewrite the function as:

$[(3x^3+4)^2]^{1/3}$ or $(3x^3+4)^{\frac{2}{3}}$ .

Then, apply the chain rule (first multiply the exponent by the coefficient in front of the binomial [1], then decrease the exponent of the binomial by 1, and finally take the derivative of the binomial):

$\frac{2}{3}(3x^3+4)^{-1/3}(9x^2)$

When simplifiying, change negative exponents to positive ones. Therefore, the answer is:

$f{}'(x)=\frac{6x^2}{\sqrt[3]{3x^3+4}}$ .

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Example Question #15 : Understanding The Limiting Process.

If $y=x^{3}e^{x}$ , then $\frac{dy}{dx} = ?$

Possible Answers:

$3x+e^{x}$

$x^{4}e$

$x^{2}e^{x}(3+x)$

$3x^{2}e^{x}$

$3x^{4}2e^{x}$

Correct answer:

$x^{2}e^{x}(3+x)$

Explanation:

The correct answer is $x^{2}e^{x}(3+x)$ .

We must use the product rule to solve. Remember that the derivative of $e^{x}$ is $e^{x}$ .

$y'=(3x^{2})(e^{x})+(x^{3})(e^{x})$

$y'=x^{2}e^{x}(3+x)$

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Example Question #52 : Limits Of Functions (Including One Sided Limits)

Differentiate $y=e^{x}+x^e$ .

Possible Answers:

$y'=e^{x^2}+x^{e^2}$

$y' = e^x+ex^{e-1}$

y'=e^x+1

y'=e^x

Correct answer:

$y' = e^x+ex^{e-1}$

Explanation:

The derivative of e^x is equal to e^x therefore the first part of the equation remains the same.

The second part requires regular differential rules.

$x^ndx=nx^{n-1}$

Therefore when differentiating x^e you get $ex^{e-1}$ .

Combining the first and second part we get the final derivative:

$y'=e^x+ex^{e-1}$ .

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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #5 : Understanding The Limiting Process.

Example Question #6 : Understanding The Limiting Process.

Example Question #7 : Understanding The Limiting Process.

Example Question #181 : Functions, Graphs, And Limits

Example Question #11 : Understanding The Limiting Process.

Example Question #11 : Understanding The Limiting Process.

Example Question #11 : Understanding The Limiting Process.

Example Question #51 : Limits Of Functions (Including One Sided Limits)

Example Question #15 : Understanding The Limiting Process.

Example Question #52 : Limits Of Functions (Including One Sided Limits)

All AP Calculus AB Resources

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