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Precalculus : Find the Slope of a Line Tangent to a Curve At a Given Point

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Example Questions

Example Question #1 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Find the slope of the line y=3x^3 at the point (1,3) .

Possible Answers:

m=9

m=1

m=3

m=9x^2

Correct answer:

m=9

Explanation:

First find the slope of the tangent to the line by taking the derivative.

Using the Exponential Rule we get the following,

y=3x^3

$\frac{dy}{dx}=9x^2$ .

Then plug 1 into the equation as 1 is the point to find the slope at.

$\frac{dy}{dx}=9(1)^2=9(1)=9$ .

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Example Question #2 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Find the slope of the following expression at the point (2,3)

y=x^3+2x^2+x .

Possible Answers:

Correct answer:

Explanation:

One way of finding the slope at a given point is by finding the derivative. In this case, we can take the derivative of y with respect to x, and plug in the desired value for x.

y=x^3+2x^2+x

Using the exponential rule we get the following derivative,

$\frac{\mathrm{d}y }{\mathrm{d} x}=3x^2 +4x+1$ .

Plugging in x=2 from the point 2,3 gives us the final slope,

3(2)^2+4(2)+1

=3(4)+8+1

=12+9=21

Thus our slope at the specific point is .

Note that in this case, using the y coordinate was not necessary.

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Example Question #3 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Find the slope of the tangent line of the function at the given value

f(x)=3x^2

x=3 .

Possible Answers:

Correct answer:

Explanation:

To find the slope of the tangent line of the function at the given value, evaluate the first derivative for the given.

The first derivative is

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

and for this function

$f^{'}(x)=3(2)x^{2-1}=6x$

and

f'(3)=6(3)=18

So the slope is

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Example Question #1 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Find the slope of the tangent line of the function at the given value

f(x)=x^2+4x

x=1 .

Possible Answers:

Correct answer:

Explanation:

To find the slope of the tangent line of the function at the given value, evaluate the first derivative for the given value.

The first derivative is

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

and for this function

$f^{'}(x)=2x^{2-1}+4x^{1-1}=2x+4$

and plugging in the specific x value we get,

f'(1)=2(1)+4=6

So the slope is

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Example Question #5 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Consider the function y=2x^2+10x . What is the slope of the line tangent to the graph at the point (1,12) ?

Possible Answers:

Correct answer:

Explanation:

Calculate the derivative of y=2x^2+10x by using the derivative rules. The derivative function determines the slope at any point of the original function.

The derivative is: y'=4x+10

With the given point (1,12) , x=1 . Substitute this value to the derivative function to determine the slope at that point.

y'=4(1)+10=14

The slope of the tangent line that intersects point (1,12) is .

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Precalculus : Find the Slope of a Line Tangent to a Curve At a Given Point

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Example Question #2 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Example Question #3 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Example Question #1 : Find The Slope Of A Line Tangent To A Curve At A Given Point

Example Question #5 : Find The Slope Of A Line Tangent To A Curve At A Given Point

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