To find the equation of the tangent line to the graph of f(x) = 14x^5 - 13e^x at the point (0, -10), we need to determine the slope (m) and the y-intercept (b).
1. Slope (m) of the Tangent Line:
- The slope of the tangent line at a point on the graph of a function f(x) is given by the derivative of f(x) evaluated at that point.
- First, find the derivative of f(x):
f'(x) = 70x^4 - 13e^x
- Evaluate the derivative at x = 0 (since the point of tangency is (0, -10)):
f'(0) = 70(0)^4 - 13e^0 = 0 - 13(1) = -13
- So, the slope (m) of the tangent line is m = -13.
2. y-intercept (b) of the Tangent Line:
- We already know that the point of tangency is (0, -10).
- Since the tangent line passes through this point, we can use the point-slope form of the equation of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
- Substituting x1 = 0, y1 = -10, and m = -13, we get:
y - (-10) = -13(x - 0)
y + 10 = -13x
y = -13x - 10
So, the equation of the tangent line to the graph of f(x) at the point (0, -10) is y = -13x - 10.
