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Calculus 1 : Calculus

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Example Questions

Example Question #612 : Functions

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

Possible Answers:

(4.33,1.09)

(1.15,9.06)

(11.12,5.89)

(8.05,2.77)

(2.85,1.14)

Correct answer:

(8.05,2.77)

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 rules will be necessary:

Derivative of an exponential:

$d[a^u]=a^udu\ln(a)$

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.

Taking the partial derivatives of $f(x,y)=(1+2^{y})^{x}$ at (1,2)

$\frac{\delta f}{\delta x}=(1+2^{y})^{x}\ln(1+2^y);(1+2^{2})^{1}\ln(1+2^2)=8.05$

$\frac{\delta f}{\delta y}=x(2^{y}\ln(2))(1+2^{y})^{x-1};1(2^{2}\ln(2))(1+2^{2})^{1-1}=2.77$

The slope is (8.05,2.77)

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Example Question #422 : Other Differential Functions

Find the slope of the line tangent to the function $f(x)=\cos(4+x^{1.7})$ at x=2.2

Possible Answers:

-1.89

-2.95

2.12

-5.54

1.36

Correct answer:

-2.95

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

$d[\cos(u)]=-\sin(u)du$

Note that may represent complext functions.

Taking the derivative of the function $f(x)=\cos(4+x^{1.7})$ at x=2.2

The slope of the tangent is

$f'(x)=-1.7x^{0.7}\sin(4+x^{1.7})$

$f'(x)=-1.7(2.2)^{0.7}\sin(4+2.2^{1.7})=-2.95$

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Example Question #423 : Other Differential Functions

Find the slope of the line tangent to the function $f(x)=1.32^{x^{4.93}}$ at x=1.02

Possible Answers:

2.009

11.013

0.017

8.001

1.005

Correct answer:

2.009

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential:

$d[a^u]=a^udu\ln(a)$

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

Taking the derivative of the function $f(x)=1.32^{x^{4.93}}$ at x=1.02

The slope of the tangent is

$f'(x)=4.93x^{3.93}(1.32^{x^{4.93}})\ln(1.32)$

$f'(1.02)=4.93(1.02)^{3.93}(1.32^{1.02^{4.93}})\ln(1.32)=2.009$

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Example Question #421 : How To Find Differential Functions

Find the derivative at x=3 .

x^3+x^2

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

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

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

The derivative is 3x^2+2x.

Now, substitute for .

3(3^2)+2(3)=33

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Example Question #422 : How To Find Differential Functions

Find the derivative at x=2 .

x^2-4x

Possible Answers:

Correct answer:

Explanation:

First, find the derivative.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

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

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

The derivative is 2x-4 .

Now, substitute for .

2(2)-4=0

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Example Question #421 : Other Differential Functions

Find the derivative at x=1 .

-3x^2-2x+11

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

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

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

Recall that the derivative of a constant is zero.

$\frac{d}{dx}11=0$

Thus, the derivative is -6x-2.

Now, substitute for .

-6(1)-2=-8

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Example Question #431 : Other Differential Functions

Find the derivative.

x^4+2x-3x^3

Possible Answers:

4x^3+9x^2+2

-4x^3-9x^2+2

4x^3-9x^2+2

4x^3-9x^2-2

Correct answer:

4x^3-9x^2+2

Explanation:

Use the power rule to find the derivative.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

$\frac{d}{dx}x^4=4x^3$

$\frac{d}{dx}-3x^3=-9x^2$

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

The derivative is 4x^3-9x^2+2

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Example Question #432 : Other Differential Functions

Find the derivative at x=4 .

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

Possible Answers:

394

308

400

270

Correct answer:

394

Explanation:

First, use the power rule to find the derivative.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

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

$\frac{d}{dx}8x^3=24x^2$

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

The derivative is 24x^2+2x+2

Now, substitute for .

24(4^2)+2(4)+2

=394

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Example Question #433 : Other Differential Functions

Find the derivative at x=5 .

x^3-2x^2+9x

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

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

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

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

The derivative is 3x^2-4x+9

Now, substitute for .

3(5^2)-4(5)+9

=64

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Example Question #621 : Functions

Find the derivative.

7x^4+8x^3+9x^2+10x+1

Possible Answers:

28x^3+24x^2+28x

28x^3+24x^2+18x+10

28x^3+24x^2+18x

28x^3

Correct answer:

28x^3+24x^2+18x+10

Explanation:

Find the derivative using the power rule.

Remember the power rule:

$(x^n)'=nx^{n-1}$

We can now apply this to our situation.

$\frac{d}{dx}7x^4=28x^3$

$\frac{d}{dx}8x^3=24x^2$

$\frac{d}{dx}9x^2=18x$

$\frac{d}{dx}10x=10$

$\frac{d}{dx}1=0$

The derivative is 28x^3+24x^2+18x+10

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #612 : Functions

Example Question #422 : Other Differential Functions

Example Question #423 : Other Differential Functions

Example Question #421 : How To Find Differential Functions

Example Question #422 : How To Find Differential Functions

Example Question #421 : Other Differential Functions

Example Question #431 : Other Differential Functions

Example Question #432 : Other Differential Functions

Example Question #433 : Other Differential Functions

Example Question #621 : Functions

All Calculus 1 Resources

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