Home > Calculus > Tangent Line to a Curve

What is the equation of the line tangent to #f(x)=x ^3-3x^2 # at #x=4#?

Answer 1

Aurora Alvarado

#y=24x-80#

First we must find the slope of the tangent line at #x=4#.

#[1]" "f'(x_0)=lim_(x->x_0)(f(x)-f(x_0))/(x-x_0)#

#[2]" "f'(4)=lim_(x->4)(f(x)-f(4))/(x-4)#

#[3]" "f'(4)=lim_(x->4)((x^3-3x^2)-(4^3-3(4)^2))/(x-4)#

#[4]" "f'(4)=lim_(x->4)(x^3-3x^2-16)/(x-4)#

#[5]" "f'(4)=lim_(x->4)(cancel((x-4))(x^2+x+4))/cancel(x-4)#

#[6]" "f'(4)=lim_(x->4)(x^2+x+4)#

#[7]" "f'(4)=(4)^2+4+4#

#[8]" "f'(4)=24#

#color(blue)(m=24)#

Now we must get the point of #f(x)# at #x=4#.

#[1]" "y=x^3-3x^2#

#[2]" "y=4^3-3(4)^2#

#[3]" "y=64-48#

#[4]" "y=16#

#color(red)(P(4,16))#

We can find the equation of the tangent line using the point-slope form.

#[1]" "y-y_0=m(x-x_0)#

#[2]" "y-16=24(x-4)#

#[3]" "y-16=24x-96#

#[4]" "y=24x-96+16#

#[5]" "color(pink)(y=24x-80)#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emma Aldridge

The equation of the line tangent to f(x)=x^3-3x^2 at x=4 is y = 16x - 32.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.