1. Let f(x)= -x^3 - 2x^2 + x + 1 and g(x)= ln(x+1)+1
a. Find the equation of the line tangent to f at x=0
b. Show that g has the same tangent line as f at x=0
2. Given x=t^4 - t^2 + t and y=2t^3 - t, find dy/dx
3. Given x^2 + 3y^2 = 1
a. Find y'
b. Find the slope of the curve at the point (.5,.5)
c. Find the equation of the line tangent to the ellipse at (.5,.5)
a)
So the slope of the tangent line at x = 0 is:
The tangent line is thus: where b is the intercept. This line touches the function y = f(x), and we know that y = f(0) = 1. So the line passes through the point (0, 1). Thus
gives b = 1.
Thus your tangent line is . (See graph below.)
b) To show that g(x) has the same tangent line at x = 0, all we need to do is verify that g(0) = 1 and that g'(0) = 1.
So:
(Check!)
so
(Check!)
-Dan