BC ONLYExample 9 __
Find the equation of the tangent to F(t) = ( cos t, 2 sin^2 t ) at the point where t =
.
SOLUTION: Since = −sin t and = 4 sin t cos t, we see that
.At . The equation of the tangent is
.C. INCREASING AND DECREASING FUNCTIONS
Case I. Functions with Continuous Derivatives
A function y = f (x) is said to be on an interval if for all a and b in the
interval such that . To find intervals over which f (x) , that
is, over which the curve , analyze the signs of the derivative to determine
where .
Example 10 __
For what values of x is f (x) = x^4 − 4x^3 , increasing? decreasing?
SOLUTION: f ′(x) = 4 x 3 − 12 x 2 = 4 x 2 (x − 3).
With critical values at x = 0 and x = 3, we analyze the signs of f ′ in three
intervals:
The derivative changes sign only at x = 3. Thus,