Calculus II                                                    Quiz #2                            February 25, 2005

Name____________________              R.  Hammack                               Score ______

(1)     Find the following definite integrals. Use either the Fundamental Theorem of Calculus or the various properties of definite integrals, whichever is most applicable. Write your final answer in the simplest possible form.

(a)  ∫_1^05(1 - x^2)^(1/2)dx=  -5∫_0^1(1 - x^2)^(1/2)dx=-51/4π 1^2= -(5π)/4

(Using the fact that the graph of  y=(1 - x^2)^(1/2) is a semicircle of radius 1, centered at the origin.

(b)  ∫_ (1/e)^e 1/xdx= [ln(x)] _ (1/e)^e=ln(e)-ln(1/e)=ln(e)+ln(e)=2

(c) ∫_0^1 (x^3-x+4)dx= [x^4/4 - x^2/2 + 4x] _0^1=1/4-1/2+4=15/4

(2)    Consider the function  F(x)=∫_3^xe^t(t^2+2t-15)dt

(a)  F '(x)= e^x(x^2+2x-15)

(b)  At which x values (if any) is the tangent to the graph of  y=F(x) horizontal?

F '(x)=e^x(x^2+2x-15)=e^x(x-3)(x+5)

Derivative is 0 when x = 3 or x = -5, so that's where the tangent to F is horizontal.