post 35

We have been taking ap tests all week long. Here are some problems.


1. d/dx cos^2(x^3)

chain rule!
-2cos(x^3)sin(x^3)(3x^2)
-6x^2cos(x^3)sin(x^3)


2. An equation of the line tangent to the graph of y=cos(2x) at x=pi/4 is
tangent line steps: take derivative, plug in x to get slope, use point slope formula, and if not given y plug x value into original

cos(2x)
-2sin(2x)
-2sin2(pi/4)
-2sin(pi/2)
-2=slope

cos(2(pi/4))
cos(pi/2)=0

y=-2(x-pi/4)


3. Let f be a function defined for all real numbers x. If f^1(x)=(4-x^2)/x2, then f is decreasing on the interval *(4-x^2 is really the absolute value of 4-x^2)

first derivative test!
f^1(x)=(4-x^2)/x2 =0
4-x^2=0
-x$2=4
x=+ or - 2

(-infinity,-2) (-2,2) (2,infinity)
plug in -3=-ve/decreasing
plug in 0=-ve/decreasing
plug in 3=+ve/increasing

So decreasing on the interval (-infinity,2)


4. 0S(pi/4) e^tanx/cos^2x dx is
u=tanx du=sec^2x = 1/cos^2x
0Spi/4 e^u du
e^tanx *(have to integrate from 0 to pi/4)
e^tan(pi/4) - e^tan0
e^1 - e^0
=e-1


5. If f(x)=ln((x^2)-1), then f^1(x) *(x^2-1)is really the absolute value of x^2-1

just take derivative!
f^1(x)=ln(x^2-1)
(1/x^2-1)(2x)
2x/(x^2)-1


I'm not to great with integration!