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What is the equation of the line tangent to # f(x)=x^2 + 2x# at # x=3#?

Answer 1

Christopher Agresta

#y=8x-9#

Locate the function's point of tangency, or the location where the tangent line will intersect, first:

#f(3)=3^2+2(3)=15#

Thus, the tangent line passes through the point #(3,15)#.

To find the slope of the tangent line, find the value of the derivative at #x=3#.

Apply the power rule to determine the function's derivative.

#f(x)=x^2+2x#

#f'(x)=2x+2#

The tangent line's slope is

#f'(3)=2(3)+3=8#

So, we know that the tangent line passes through the point #(3,15)# and has a slope of #8#.

These can be related as a linear equation in point-slope form, which takes a point #(x_1,y_1)# and slope #m#:

#y-y_1=m(x-x_1)#

Consequently, the tangent line's equation is

#y-15=8(x-3)#

which one could reword as

#y=8x-9#

The original function and its tangent line are graphed:

graph{y-8x+9)(x^2+2x-y)=0 [-2, 7, -11, 40]}

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Answer 2

Emilia Aguilar

The equation of the line tangent to f(x)=x^2 + 2x at x=3 is y = 11x - 16.

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