Let f(x)=2x+15−19ex. Then the equation of the tangent line to the graph of f(x) at the point (0,−4) is given by y=mx+b for
m=?
b=?
f = 2x+15-19e^x
f' = 2-19e^x
f'(0) = 2-19 = -17
so, the tangent line is
y+4 = -17(x-0)
see the graphs at
http://www.wolframalpha.com/input/?i=plot+y%3D2x%2B15-19e%5Ex%2C+y%3D-17x-4
To find the equation of the tangent line to the graph of f(x) at the point (0, -4), we need to find the slope and the y-intercept of the line.
Step 1: Find the slope (m) of the tangent line.
The slope of the tangent line is equal to the derivative of f(x) evaluated at x = 0.
f'(x) = 2 - 19e^x (taking the derivative of 2x + 15 - 19e^x with respect to x)
f'(0) = 2 - 19e^0 (substituting x = 0)
f'(0) = 2 - 19 (since e^0 = 1)
f'(0) = -17
So, the slope (m) of the tangent line is -17.
Step 2: Find the y-intercept (b) of the tangent line.
Since the tangent line passes through the point (0, -4), we can substitute these values into the equation y = mx + b and solve for b:
-4 = -17(0) + b (substituting m = -17, x = 0, and y = -4)
-4 = 0 + b
b = -4
So, the y-intercept (b) of the tangent line is -4.
Therefore, the equation of the tangent line to the graph of f(x) at the point (0, -4) is y = -17x - 4.
To find the equation of the tangent line to the graph of f(x) at the point (0, -4), we need to determine the slope (m) and the y-intercept (b) of the line.
1. Find the slope (m):
The slope of a tangent line to a curve at a given point can be found by taking the derivative of the function at that point. So, let's find the derivative of f(x):
f(x) = 2x + 15 - 19e^x
To find the derivative, we differentiate each term separately:
f'(x) = (d/dx)(2x) + (d/dx)(15) - (d/dx)(19e^x)
The derivative of 2x with respect to x is 2.
The derivative of 15 with respect to x is 0 because it's a constant.
The derivative of 19e^x with respect to x is 19e^x.
Therefore, the derivative f'(x) is:
f'(x) = 2 + 0 - 19e^x
= 2 - 19e^x
Now, substitute x = 0 into f'(x) to find the slope at the point (0, -4):
m = f'(0) = 2 - 19e^0
= 2 - 19(1)
= 2 - 19
= -17
So, the slope (m) of the tangent line is -17.
2. Find the y-intercept (b):
We have the point (0, -4) which lies on the tangent line. The general equation of a line is y = mx + b, where b is the y-intercept. Substitute the values from the point into the equation:
-4 = (-17)(0) + b
-4 = b
Therefore, the y-intercept (b) is -4.
So, the equation of the tangent line to the graph of f(x) at the point (0, -4) is:
y = -17x - 4
Hence, the values for m and b are:
m = -17
b = -4