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Calculus 1 : Calculus

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Example Questions

Example Question #314 : How To Find Differential Functions

Find the derivative.

$\cos(x)\sin (x)$

Possible Answers:

$\cos (x)\sin (x)$

$\cos^2 (x)-\sin^2(x)$

$\cos ^{2}(x)$

Correct answer:

$\cos^2 (x)-\sin^2(x)$

Explanation:

Use the product rule to find the derivative.

Recall that the derivative of $\sin$ is $\cos$ , and the derivative of $\cos$ is $-\sin$ .

$\cos (x)\cos (x)-\sin (x)\sin (x)$

Simplify.

$\cos^2 (x)-\sin^2(x)$

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Example Question #1532 : Calculus

Find the slope of the tangent line at x=8 .

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

Possible Answers:

402

371

330

365

Correct answer:

371

Explanation:

First, find the derivative by using the power rule.

$\frac{d}{dx}2x^3=6x^2$

$\frac{d}{dx}-x^{2}=-2x$

$\frac{d}{dx}3x=3$

Thus, the derivative is $6x^{2}-2x+3$ .

Now, substitute for .

6(8)^2-2(8)+3=371

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Example Question #1531 : Calculus

Find the derivative.

2x-5

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}2x=2$

$\frac{d}{dx}-5=0$

Thus, the derivative equals .

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Example Question #316 : How To Find Differential Functions

Find the derivative at x=2.5 .

4x+17

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

$\frac{d}{dx}4x=4$

$\frac{d}{dx}17=0$

The derivative is , regardless of the value.

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Example Question #502 : Functions

Find the derivative at x=4 .

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

Possible Answers:

$-\frac{1}{256}$

$-\frac{9}{256}$

$-\frac{17}{256}$

$-\frac{13}{256}$

Correct answer:

$-\frac{9}{256}$

Explanation:

Use the quotient rule to find the derivative.

$\frac{x^{3}(0)-3(3x^{2})}{x^{6}}$

$-\frac{9x^{2}}{x^{6}}=-\frac{9}{x^{4}}$

Now, substitute 4 for x.

$-\frac{9}{4^4}=-\frac{9}{256}$

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Example Question #1532 : Calculus

Find the derivative.

9-3x

Possible Answers:

-3x

Correct answer:

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}9=0$

$\frac{d}{dx}-3x=-3$

The derivative is .

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Example Question #503 : Differential Functions

Find the derivative at x=6 .

$2x^{2}-6x$

Possible Answers:

Correct answer:

Explanation:

Begin by finding the derivative using the power rule.

$\frac{d}{dx}2x^{2}=4x$

$\frac{d}{dx}-6x=-6$

The derivative is 4x-6.

Now, substitute for .

4(6)-6=24-6=18

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Example Question #321 : How To Find Differential Functions

Find the slope of the function f(x,y)=(x^2+y)^3 at the point (2,5) .

Possible Answers:

(81,27)

(3,1)

(243,81)

(9,3)

(972,243)

Correct answer:

(972,243)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y)=(x^2+y)^3 at the point (2,5)

$\frac{\delta f}{\delta x}=6x(x^2+y)^2;6(2)(2^2+5)^2=972$

$\frac{\delta f}{\delta y}=3(x^2+y)^2;3(2^2+5)^2=243$

The slope is (972,243)

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Example Question #321 : How To Find Differential Functions

Find the slope of the function f(x,y,z)=1^x+2^y+3^z at the point (3,2,1) .

Possible Answers:

(1,4,3)

(8.1,0,9.7)

(0,4.4,6.6)

(0,2.8,3.3)

(1.1,2.2,3.3)

Correct answer:

(0,2.8,3.3)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y,z)=1^x+2^y+3^z at the point

$\frac{\delta f}{\delta x}=1^xln(1);1^3ln(1)=0$

$\frac{\delta f}{\delta y}=2^yln(2);2^2ln(2)=2.8$

$\frac{\delta f}{\delta z}=3^zln(3);3^1ln(3)=3.3$

The slope is (0,2.8,3.3)

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Example Question #1533 : Calculus

Find the slope of the function $f(x,y,z)=2^{(x+y+z)^2}$ at the point (0,1,2) .

Possible Answers:

(603.3,603.3,603.3)

(903.3,602.2,301.1)

(2129.3,2129.3,2129.3)

(212.3,424.6,636.9)

(709.8,1419.5,2129.3)

Correct answer:

(2129.3,2129.3,2129.3)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y,z)=2^{(x+y+z)^2}$ at the point (0,1,2)

$\frac{\delta f}{\delta x}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

$\frac{\delta f}{\delta y}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

$\frac{\delta f}{\delta z}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

The slope is (2129.3,2129.3,2129.3)

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #314 : How To Find Differential Functions

Example Question #1532 : Calculus

Example Question #1531 : Calculus

Example Question #316 : How To Find Differential Functions

Example Question #502 : Functions

Example Question #1532 : Calculus

Example Question #503 : Differential Functions

Example Question #321 : How To Find Differential Functions

Example Question #321 : How To Find Differential Functions

Example Question #1533 : Calculus

All Calculus 1 Resources

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