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AP Calculus AB : Understanding the limiting process.

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Example Questions

Example Question #1 : Understanding The Limiting Process.

Find the derivative.

y = sec (5x³)

Possible Answers:

y' = –csc(5x³)cot(5x³)(15x²)

y' = –sec(5x³)tan(5x³)(15x²)

y' = sec(5x³)tan(5x³)(15x²)

y' = –csc(5x³)cot(5x³)

y' = sec(5x³)tan(5x³)

Correct answer:

y' = sec(5x³)tan(5x³)(15x²)

Explanation:

The derivative of the function y = sec(x) is sec(x)tan(x). First take the derivative of the outside of the function: y = sec(4x³) : y' = sec(5x³)tan(5x³). Then take the derivative of the inside of the function: 5x³ becomes 15x². So your final answer is: y' = ec(5x³)tan(5x³)15x²

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Example Question #1 : Understanding The Limiting Process.

Find the slope of the tangent line to the graph of f at x = 9, given that f(x) = –x²+ 5√(x)

Possible Answers:

–18 + (5/6)

–18

–18 – (5/6)

18 + (5/6)

Correct answer:

–18 + (5/6)

Explanation:

First find the derivative of the function.

f(x) = –x² + 5√(x)

f'(x) = –2x + 5(1/2)x^–1/2

Simplify the problem

f'(x) = –2x + (5/2x^1/2)

Plug in 9.

f'(3) = –2(9) + (5/2(9)^1/2)

= –18 + 5/(6)

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Example Question #2 : Understanding The Limiting Process.

Find the derivative

(x + 1)/(x – 1)

Possible Answers:

(–2)/(x + 1)²

(x + 1) + (x – 1)

(–2)/(x – 1)²

(–2)/(x – 1)

Correct answer:

(–2)/(x – 1)²

Explanation:

Rewrite problem.

(x + 1)/(x – 1)

Use quotient rule to solve this derivative.

((x – 1)(1) – (x + 1)(1))/(x – 1)²

(x – 1) – x – 1)/(x – 1)²

–2/(x – 1)²

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Example Question #35 : Limits Of Functions (Including One Sided Limits)

$\frac{d}{dx}sin^{-1} (3x)$

Possible Answers:

$\frac{3x}{\sqrt{1-9 x^2}}$

$\frac{3}{\sqrt{1-9 x^2}}$

$\frac{3x}{\sqrt{1- x^2}}$

$\frac{3}{\sqrt{1-3x^2}}$

Correct answer:

$\frac{3}{\sqrt{1-9 x^2}}$

Explanation:

Use the chain rule and the formula

$\frac{d}{dx}sin^{-1}u = \frac{1}{\sqrt{1-x^2}}$

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Example Question #4 : Understanding The Limiting Process.

Find the derivative of

$f(x)=4(x^{3}-2x+11)(x-2)$

Possible Answers:

$4x^{4}-8x^{3}-8x^{2}+60x-22$

24x-2

$f'(x)=16x^{3}-24x^{2}-16x+60$

$4(3x^{2}-2)$

$3x^{2}-8x+15$

Correct answer:

$f'(x)=16x^{3}-24x^{2}-16x+60$

Explanation:

The answer is $f'(x)=16x^{3}-24x^{2}-16x+60$ . It is easy to solve if we multiply everything together first before taking the derivative.

$f(x)=4(x^{3}-2x+11)(x-2)$

$f(x)=4(x^{4}-2x^{3}+-2x^{2}+4x+11x-22)$

$f(x)=4x^{4}-8x^{3}-8x^{2}+60x-22$

$f'(x)=16x^{3}-24x^{2}-16x+60$

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Example Question #3 : Understanding The Limiting Process.

Differentiate f(x)=5x^3 .

Possible Answers:

15x^2

15x^3

15x^4

(5/3)x^2

8x^2

Correct answer:

15x^2

Explanation:

Using the power rule, multiply the coefficient by the power and subtract the power by 1.

f(x)=5x^3

$f'(x)=5*3(x^{3-1})$

f'(x)=15x^2

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Example Question #4 : Understanding The Limiting Process.

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

Possible Answers:

$\frac{-2}{x}$

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

$\frac{1}{x^3}$

2ln(x^2)

ln(x^2)

Correct answer:

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

Explanation:

Use the product rule:

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

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

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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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AP Calculus AB : Understanding the limiting process.

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Understanding The Limiting Process.

Example Question #1 : Understanding The Limiting Process.

Example Question #2 : Understanding The Limiting Process.

Example Question #35 : Limits Of Functions (Including One Sided Limits)

Example Question #4 : Understanding The Limiting Process.

Example Question #3 : Understanding The Limiting Process.

Example Question #4 : Understanding The Limiting Process.

Example Question #5 : Understanding The Limiting Process.

Example Question #6 : Understanding The Limiting Process.

Example Question #7 : Understanding The Limiting Process.

All AP Calculus AB Resources

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