1. Find the slope m and an equation of the tangent line to the graph of the function f a the point (2, 38)
f(x)=9x -2x +6
To find the slope and equation of the tangent line to the graph of a function at a given point, you can follow these steps:
Step 1: Find the derivative of the function to determine its slope at any point. For the function f(x) = 9x - 2x^2 + 6, take its derivative using the power rule for derivatives:
f'(x) = d/dx (9x - 2x^2 + 6)
= 9 - 4x
Step 2: Substitute the x-coordinate of the point (2, 38) into the derivative you just found to calculate the slope at that point:
m = f'(2) = 9 - 4(2) = 9 - 8 = 1
So, the slope of the tangent line to the graph of the function f at the point (2, 38) is 1.
Step 3: Use the point-slope form of a line to write the equation of the tangent line. The point-slope form is:
y - y1 = m(x - x1)
where m is the slope, and (x1, y1) represents the point (2, 38). Plugging in the values, we have:
y - 38 = 1(x - 2)
y - 38 = x - 2
Simplifying the equation, we get:
y = x + 36
Hence, the equation of the tangent line to the graph of the function f at the point (2, 38) is y = x + 36.