Homework Soursefx5ex AFind the slope of the graph of fx at the point where
f(x)=5-e^x
A.Find the slope of the graph of f(x) at the point where the graph crosses the x-axis.
slope = ??
B. Find the equation of the tangent line to the curve at this point.
y=????
C.ind the equation of the line perpendicular to the tangent line at this point. (This is the normal line.)
y=???
A.Find the slope of the graph of f(x) at the point where the graph crosses the x-axis.
slope = ??
B. Find the equation of the tangent line to the curve at this point.
y=????
C.ind the equation of the line perpendicular to the tangent line at this point. (This is the normal line.)
y=???
Solution
f(x) = 5 - e^x
f\'(x) = -e^x
(A) For crossing the x-axis, f(x) = 0
5 - e^x = 0
e^x = 5
x = ln5
Slope at x = ln5, f \' (ln5) = -e^(ln5) = -5
(B) At x = ln5, y = 0
Slope = -5
Equation of tangent :
y - 0 = -5 (x - ln5)
y = -5x + 5ln5
5x + y = 5 ln5 (Tangent)
(C) Slope of the normal, m = -1/f \' (x)
At x = ln5, f\'(x) = -5
slope of normal = 1/5
Equation of normal :
y - 0 = (1/5)(x - ln5)
5y = x - ln5
x - 5y = ln5 (Normal)
