for the function f(x)=4x^2 +7, find the equation of the tangent line to the graph of f at x+1. if the equation of the tangent line is written as y=mx+b. m=? b=?
To find the equation of the tangent line to the graph of f(x) at x = -1, we need to find the derivative of f(x) and evaluate it at x = -1.
Step 1: Find the derivative of f(x)
To find the derivative of f(x), we can use the power rule for differentiation. For any function of the form f(x) = ax^n, the derivative is given by f'(x) = n*ax^(n-1).
For f(x) = 4x^2 + 7, the derivative f'(x) is:
f'(x) = 2 * 4x^(2-1) = 8x
Step 2: Evaluate the derivative at x = -1
To find the slope of the tangent line, we need to evaluate the derivative at x = -1.
f'(-1) = 8 * (-1) = -8
Step 3: Find the equation of the tangent line
The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.
Since we know the slope (m = -8), we just need to find the y-intercept (b). To do that, we can substitute the point (-1, f(-1)) into the equation y = mx + b.
f(-1) = 4(-1)^2 + 7 = 4 + 7 = 11
Substituting the values into the equation:
11 = -8 * (-1) + b
Simplifying:
11 = 8 + b
b = 11 - 8
b = 3
Therefore, the equation of the tangent line is y = -8x + 3 and the values of m and b are:
m = -8
b = 3