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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #106 : Derivative Review

What is the slope of a function $f(x)=4x^{2}+x-11$ at the point (-2,5) ?

Possible Answers:

Correct answer:

Explanation:

By definition, slope is the first derivative of a given function f(x) .

Since $f(x)=4x^{2}+x-11$ here, we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)4x^{2-1}+(1)x^{1-1}-(0)11=8x+1$ .

At the point (-2,5) , we have x=-2 , and so

f'(-2)=8(-2)+1=-16+1=-15 .

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Example Question #21 : Derivative At A Point

What is the slope of a function $f(x)=7x^{2}-2x+10$ at the point (-1,4) ?

Possible Answers:

None of the above

Correct answer:

Explanation:

By definition, slope is the first derivative of a given function f(x) .

Since $f(x)=7x^{2}-2x+10$ here, we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)7x^{2-1}-(1)2x^{1-1}+(0)10=14x-2$ .

At the point (-1,4) , we have x=-1 , and so

f'(-1)=14(-1)-2=-14-2=-16 .

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Example Question #108 : Derivative Review

Given a function $f(x)=9x^{2}-12x+20$ , what is its slope at the point (4,5) ?

Possible Answers:

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at a given point.

Given

$f(x)=9x^{2}-12x+20$ . we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)9x^{2-1}-(1)12x^{(1-1)}+(0)20=18x-12$ .

Since the -coordinate of (4,5) is , the slope

f'(4)=18(4)-12=72-12=60 .

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Example Question #23 : Derivative At A Point

Given a function $f(x)=-3x^{2}+14x-23$ , what is its slope at the point (-2,-2) ?

Possible Answers:

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at a given point.

Given

$f(x)=-3x^{2}+14x-23$ . we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=-3(2)x^{2-1}+(1)14x^{1-1}-(0)23=-6x+14$ .

Since the -coordinate of (-2,-2) is , the slope

f'(-2)=-6(-2)+14=12+14=26 .

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Example Question #110 : Derivative Review

Given a function $f(x)=\frac{1}{x^{2}}-3x+2$ , what is its slope at the point (4,1) ?

Possible Answers:

None of the above.

$\frac{32}{97}$

$-\frac{32}{97}$

$\frac{97}{32}$

$-\frac{97}{32}$

Correct answer:

$-\frac{97}{32}$

Explanation:

Slope is defined as the first derivative of a function at a given point. Given $f(x)=\frac{1}{x^{2}}-3x+2$ or $f(x)=x^{-2}-3x+2$ . we can use the Power Rule ( $\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ ) to derive $f'(x)=(-2))x^{-2-1}-(1)3x^{1-1}+(0)2=-2x^{-3}-3=-\frac{2}{x^{3}}-3$ . Since the -coordinate of (4,1) is , the slope $f'(4)=-\frac{2}{4^{3}}-3=-\frac{2}{64}-3=-\frac{1}{32}-3=-\frac{1}{32}-\frac{96}{32}=-\frac{97}{32}$ .

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Example Question #22 : Derivative At A Point

Find the derivative of $y=\left | x-3\right |$ at x=3 .

Possible Answers:

$\pm1$

Does not exist.

Correct answer:

Does not exist.

Explanation:

Split the absolute value into both positive and negative components.

$y=\left | x-3\right | = \pm(x-3)$

Take their derivatives.

$\frac{d}{dx}[y=x-3]=1$

$\frac{d}{dx}[y=-(x-3)]=-1$

At x=3 , there exists a spike in the graph. For spikes, the derivative does not exist under this exception.

The answer is:

$\textup{Does not exist.}$

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Example Question #111 : Derivative Review

What is the slope of a function $f(x)=10x^{2}-25x+12$ at the point (5,7) ?

Possible Answers:

100

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at given point.

We are given the function

$f(x)=10x^{2}-25x+12$ and a point (5,7) , so we need to find the derivative f'(x) and solve for the point's -coordinate.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all nonzero , we can derive

$f'(x)=(2)10x^{2-1}-(1)25x^{1-1}+(0)12=20x-25$ .

Substituting the -coordinate , we have a slope:

f'(5)=20(5)-25=100-25=75 .

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Example Question #112 : Derivative Review

What is the slope of a function $f(x)=x^{2}+7x-3$ at the point (-3,0) ?

Possible Answers:

$\frac{1}{3}$

None of the above

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at given point.

We are given the function

$f(x)=x^{2}+7x-3$ and a point (-3,0) , so we need to find the derivative f'(x) and solve for the point's -coordinate.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all nonzero , we can derive

$f'(x)=(2)x^{2-1}+(1)7x^{1-1}-(0)3=2x+7$ .

Substituting the -coordinate , we have a slope:

f'(-3)=2(-3)+7=-6+7=1 .

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Example Question #111 : Derivative Review

What is the slope of a function $f(x)=6x^{2}-12x+9$ at the point (-10,10) ?

Possible Answers:

None of the above

132

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at given point.

We are given the function

$f(x)=6x^{2}-12x+9$ and a point (-10,10) , so we need to find the derivative f'(x) and solve for the point's -coordinate.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all nonzero , we can derive

$f'(x)=(2)6x^{2-1}-(1)12x^{1-1}+(0)9=12x-12$ .

Substituting the -coordinate , we have a slope:

f'(-10)=12(-10)-12=-120-12=-132 .

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Example Question #112 : Derivative Review

What is the slope of $f(x)=4x^{2}-5x+12$ at (1,-1) ?

Possible Answers:

Correct answer:

Explanation:

Slope is defined as the first derivative of a given function.

Since,

$f(x)=4x^{2}-5x+12$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)4x^{2-1}-(1)5x^{1-1}+(0)12=8x-5$ .

At the point (1,-1) , the -coordinate is .

Thus, the slope is

f'(1)=8(1)-5=8-5=3 .

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #106 : Derivative Review

Example Question #21 : Derivative At A Point

Example Question #108 : Derivative Review

Example Question #23 : Derivative At A Point

Example Question #110 : Derivative Review

Example Question #22 : Derivative At A Point

Example Question #111 : Derivative Review

Example Question #112 : Derivative Review

Example Question #111 : Derivative Review

Example Question #112 : Derivative Review

All Calculus 2 Resources

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