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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #1571 : Calculus

Find the slope of the line normal to the function $f(x)=e^{xcos(x)}$ at the point $x=\frac{\pi}{4}$ .

Possible Answers:

-0.451

3.781

2.218

0.264

-3.781

Correct answer:

-3.781

Explanation:

The first step to finding the slope of the line normal to a point is to find the slope of the tangent at this point.

The slope of this tangent, in turn, is found by finding the value of the derivative of the function at this point.

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

Derivative of an exponential:

d[e^u]=e^udu

Trigonometric derivative:

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

Product rule: d[uv]=udv+vdu

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

Evaluating the function $f(x)=e^{xcos(x)}$ at the point $x=\frac{\pi}{4}$

The slope of the tangent is

$f'(x)=(cos(x)-xsin(x))e^{xcos(x)}$

$f'(\frac{\pi}{4})=(cos(\frac{\pi}{4})-(\frac{\pi}{4})sin(\frac{\pi}{4}))e^{(\frac{\pi}{4})cos(\frac{\pi}{4})}$

$f'(\frac{\pi}{4})=0.264$

The slope of the normal line is the negative reciprocal of this value. Thus for this problem the normal is

-3.781

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Example Question #541 : Functions

Find the derivative.

8x^2+3x-15

Possible Answers:

16x-12

16x+3

x^2+x

Correct answer:

16x+3

Explanation:

Use the power rule to find this derivative.

Remember the power rule is:

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

Now lets apply this to our problem.

$\frac{d}{dx}8x^2=16x$

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

Recall that the derivative of a constant is zero.

$\frac{d}{dx}-15=0$

Thus, the derivative is 16x+3 .

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Example Question #355 : How To Find Differential Functions

Find the derivative at x=9 .

4x^2+6x-2

Possible Answers:

Correct answer:

Explanation:

First, use the power rule to find the derivative.

Remember the power rule is:

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

Now lets apply this to our problem.

$\frac{d}{dx}4x^2=8x$

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

Recall that the derivative of a constant is zero.

$\frac{d}{dx}-2=0$

Thus, the derivative is 8x+6.

Now, substitute for .

8(9)+6=72+6=78.

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Example Question #544 : Differential Functions

Find the derivative.

2x^3-6x^2+3x

Possible Answers:

6x^2+12x+3

3x^2-12x+3

4x^2-12x+3

6x^2-12x+3

Correct answer:

6x^2-12x+3

Explanation:

Use the power rule to find this derivative.

Remember the power rule is:

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

Now lets apply this to our problem.

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

$\frac{d}{dx}-6x^2=-12x$

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

Thus, the derivative is 6x^2-12x+3 .

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Example Question #545 : Differential Functions

Find the derivative at x=1 .

3x^2-2x-1

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Remember the power rule is:

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

Now lets apply this to our problem.

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

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

Recall that the derivative of a constant is zero.

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

The derivative is 6x-2 .

Now, substitute for .

6(1)-2=4

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Example Question #1571 : Calculus

Find the derivative.

3x^7+2x^4-x^3+x^2+2

Possible Answers:

21x^6+8x^3+3x^2+2x

21x+8x^3-3x^2+2x

21x^6-8x^3+3x^2-2x

21x^6+8x^3-3x^2+2x

Correct answer:

21x^6+8x^3-3x^2+2x

Explanation:

Use the power rule to find this derivative.

Remember the power rule is:

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

Now lets apply this to our problem.

21x^6+8x^3-3x^2+2x

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Example Question #361 : How To Find Differential Functions

Find the derivative.

3x^3+2x^2+x+1

Possible Answers:

3x^2+4x+1

x^2+x+1

9x^2+4x+1

Correct answer:

9x^2+4x+1

Explanation:

Use the power rule to find the derivative.

Remember the power rule is:

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

Now lets apply this to our problem.

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

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

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

Recall that the derivative of a constant is zero.

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

Thus, the derivative is 9x^2+4x+1

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Example Question #362 : How To Find Differential Functions

Find the derivative at x=5 .

x^4

Possible Answers:

500

550

555

505

Correct answer:

500

Explanation:

Find the derivative using the power rule.

Remember the power rule is:

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

Now lets apply this to our problem.

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

Now, substitute for .

4(5)^3=500

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Example Question #1576 : Calculus

Find the derivative at x=2 .

3x^2+7x-7

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Remember the power rule is:

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

Now lets apply this to our problem.

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

$\frac{d}{dx}7x=7$

$\frac{d}{dx}-7=0$

Thus, the derivative is 6x+7 . Now, substitute for .

6(2)+7=19

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Example Question #1577 : Calculus

Find the derivative.

$\frac{4}{x^2}$

Possible Answers:

$-\frac{8}{x^3}$

$-\frac{8}{x^4}$

$-\frac{4}{x^3}$

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

Correct answer:

$-\frac{8}{x^3}$

Explanation:

Use the quotient rule to find the derivative.

Remember the quotient rule is:

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

Now lets apply this to our problem.

$\frac{x^2(0)-4(2x)}{x^4}$

$-\frac{8x}{x^4}$

$-\frac{8}{x^3}$ is the derivative.

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1571 : Calculus

Example Question #541 : Functions

Example Question #355 : How To Find Differential Functions

Example Question #544 : Differential Functions

Example Question #545 : Differential Functions

Example Question #1571 : Calculus

Example Question #361 : How To Find Differential Functions

Example Question #362 : How To Find Differential Functions

Example Question #1576 : Calculus

Example Question #1577 : Calculus

All Calculus 1 Resources

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