What is the equation of the line tangent to # f(x)=(3x-1)(x+4) # at # x=3 #?

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Answer 1 · 2016-09-06T05:25:54

#y = 29 x - 31#

We have: #f(x) = (3 x - 1) (x + 4)#

First, let's evaluate this function at the specified value of #x#:

#=> f(3) = (3 (3) - 1) ((3) + 4)#

#=> f(3) = (8) (7)#

#=> f(3) = 56#

Then, let's evaluate the derivative of the function using the "product rule":

#=> f'(x) = (3) cdot (x + 4) + (1) cdot (3 x - 1)#

#=> f'(x) = 3x + 12 + 3 x - 1#

#=> f'(x) = 6 x + 11#

Now, let's evaluate the gradient of the tangent by using the value of #x#:

#=># Gradient #= 6 (3) + 11#

#=># Gradient #= 18 + 11#

#=># Gradient #= 29#

We can now determine the equation of the line of the tangent.

The equation of a line is in the form:

#y - y_(1) = m (x - x_(1))#

Let's substitute all the required values to get:

#=> y - 56 = 29 (x - 3)#

Finally, let's solve for #y#:

#=> y = 29 x - 87 + 56#

#=> y = 29 x - 31#