How do you find the equation of the line tangent to the graph of #f(x)=x^4# at x=-1?

Question

Ezra Adams · Accepted Answer

#y=-4x-5#

#f(x)=x ^4# #f'(x)=4x^3#

#f'# ,the first derivative, is the gradient function, and gives the gradient at any point on the curve. So at the point when #x # is -1 the gradient is -4 because #4*(-1)^3#=-4

So the equation of the tangent is #y=-4x+c#

We also know that at the point on the curve where x =--1, y=1 This point is on the tangent.

1=-4+c gives c=-5 Now sketch the curve and you will see that it makes sense

Charles Anctil · Answer

undefined To find the equation of the line tangent to the graph of f(x)=x^4 at x=-1, we need to find the slope of the tangent line and the point of tangency.

To find the slope, we take the derivative of f(x) with respect to x. The derivative of f(x)=x^4 is f'(x)=4x^3.

Next, we substitute x=-1 into f'(x) to find the slope at x=-1. f'(-1)=4(-1)^3=-4.

So, the slope of the tangent line at x=-1 is -4.

To find the point of tangency, we substitute x=-1 into f(x). f(-1)=(-1)^4=1.

Therefore, the point of tangency is (-1, 1).

Using the slope-intercept form of a line, y=mx+b, where m is the slope and b is the y-intercept, we can substitute the values we found.

The equation of the line tangent to the graph of f(x)=x^4 at x=-1 is y=-4x-3.