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Let f(x) = 6x^2 - 4x(a) Find the derivative of the line tangent to the graph of f at x = 2.Slope at x = 2: ___(b) Find an equation of the line tangent to the graph of f at x= 2.Tangent line: y= ___

Sagot :

Explanation

We are given the function

[tex]f(x)=6x^2-4x[/tex]

for part A

we are to find the derivative of the line tangent to the graph of f at x = 2

So, we will have

[tex]f^{\prime}(x)=12x-4[/tex]

when x =2,

[tex]f^{\prime}(x)=12(2)-4=24-4=20[/tex]

Thus, the slope at x is 20

For part B

To get the equation of the line tangent to the graph at x =2, we will substitute x =2 into f(x)

[tex]f(2)=6(2)^2-4(2)=6\times4-8=24-8=16[/tex]

Thus, using the formula

[tex]\begin{gathered} y=mx+c \\ where \\ y=16,\text{ m=20,x=2} \\ 16=20\times2+c \\ 16=40+c \\ c=16-40 \\ c=-24 \end{gathered}[/tex]

Therefore, the equation is

[tex]y=20x-24[/tex]

Thus, the tangent line is: y= 20x -24

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