Find the slope m of the tangent line to the graph of the function at the given point and determine an equation of the tangent line.
f(x) = 2 x - 4 x^2( at ) (-2,-20)
m=
y=
take the derivative
sub in the x value of the given point to get the slope m
Now use the method you probably learned in grade 9 to find the equation of a line given the
slope and a point on it.
To find the slope of the tangent line to the graph of the function at a given point, we need to find the derivative of the function at that point. The derivative gives us the rate at which the function is changing at a particular point.
First, let's find the derivative of the function f(x) = 2x - 4x^2. To do this, we can use the power rule for differentiation. The power rule states that if we have a term of the form x^n, the derivative is given by d/dx(x^n) = n*x^(n-1).
Applying the power rule to the function f(x) = 2x - 4x^2, we get:
f'(x) = d/dx(2x) - d/dx(4x^2)
= 2 - 8x
Now that we have the derivative function f'(x), we can evaluate it at the given point (-2,-20). Substitute x = -2 into f'(x):
f'(-2) = 2 - 8*(-2)
= 2 + 16
= 18
Therefore, the slope (m) of the tangent line to the graph of the function at the point (-2,-20) is 18.
To determine an equation of the tangent line, we can use the point-slope form of a linear equation: y - y1 = m(x - x1). Plug in the values of the point and slope:
y - (-20) = 18(x - (-2))
Simplifying this equation, we get:
y + 20 = 18(x + 2)
y + 20 = 18x + 36
y = 18x + 16
Hence, the equation of the tangent line to the graph of the function f(x) = 2x - 4x^2 at the point (-2,-20) is y = 18x + 16.