AP Calculus AB : Understanding the limiting process.

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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Example Question #1 : Understanding The Limiting Process.

Find the derivative. 

y = sec (5x3)

 

 

Possible Answers:

y' = sec(5x3)tan(5x3)

y' = –csc(5x3)cot(5x3)

y' = sec(5x3)tan(5x3)(15x2)

y' = –sec(5x3)tan(5x3)(15x2)

y' = –csc(5x3)cot(5x3)(15x2)

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

Example Question #2 : Understanding The Limiting Process.

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

Possible Answers:

–18 + (5/6)

18

18 + (5/6)

–18 – (5/6)

–18

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 + (5/2x1/2)

Plug in 9. 

f'(3) = –2(9) + (5/2(9)1/2)

= –18 + 5/(6)

Example Question #1 : Understanding The Limiting Process.

Find the derivative

(x + 1)/(x – 1)

Possible Answers:

(x + 1) + (x – 1)

1

(–2)/(x + 1)2

(–2)/(x – 1)

(–2)/(x – 1)2

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

Example Question #2 : Understanding The Limiting Process.

Possible Answers:

Correct answer:

Explanation:

Use the chain rule and the formula

Example Question #4 : Understanding The Limiting Process.

Find the derivative of 

Possible Answers:

Correct answer:

Explanation:

The answer is . It is easy to solve if we multiply everything together first before taking the derivative. 

Example Question #3 : Understanding The Limiting Process.

Differentiate .

Possible Answers:

Correct answer:

Explanation:

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

Example Question #4 : Understanding The Limiting Process.

Differentiate .

Possible Answers:

Correct answer:

Explanation:

Use the product rule: 

 

Example Question #7 : Understanding The Limiting Process.

Differentiate:

Possible Answers:

Correct answer:

Explanation:

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

Example Question #5 : Understanding The Limiting Process.

Differentiate:

Possible Answers:

Correct answer:

Explanation:

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

Example Question #9 : Understanding The Limiting Process.

Find the second derivative of .

Possible Answers:

Correct answer:

Explanation:

Factoring out an x gives you .

 

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