What is the equation of the tangent line of $f (x) = (2 x + 1) (x + 2)$ $f(x)=(2x+1)(x+2)$ at $x = 2$ $x=2$ ?

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Answer 1 · 2016-02-10T02:17:28

$y = 13 x - 6$ $y= 13x - 6$

First, simplify the function through distribution so we can differentiate it easier.

$f (x) = 2 x^{2} + 5 x + 2$ $f(x)=2x^2+5x+2$

We should find the point of tangency:

$f (2) = 2 (4) + 5 (2) + 2 = 20$ $f(2)=2(4)+5(2)+2=20$

The tangent line will pass through the point $(2, 20)$ $(2,20)$ .

Through the power rule, we know that

$f'(x)=4x+5$

The slope of the tangent line will be equal to the value of the derivative at $x=2$ , or

$f'(2)=4(2)+5=13$

We know the tangent line has a slope of $13$ and passes through the point $(2,20)$ .

We can write this as an equation in $y=mx+b$ form. So far, we know that $m=13$ .

$y=13x+b$

Substitute in $(2,20)$ for $(x,y)$ .

$20=13(2)+b$

$b=-6$

Thus, the equation of the tangent line is

$y=13x-6$

Graphed are $f(x)$ and its tangent line:

graph{((2x+1)(x+2)-y)(y-13x+6)=0 [-4, 6, -10, 50]}

What is the equation of the tangent line of f(x)=(2x+1)(x+2) f(x)=(2x+1)(x+2)f(x)=(2x+1)(x+2) at x=2x=2x=2?