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Calculus  II                                                          Test #1                                                  March 3, 2004

Name____________________                     R.  Hammack                                                Score ______
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(1) Find the following antiderivatives.

(a) ∫ (1 + x^3) dx =

(b) ∫ (cos(x) + 3sec^2(x)) dx =

(c) ∫ (e^x + π) dx =

(d) ∫x/(x^4 + 1) dx =

(e) ∫ (x^4 + 1)/xdx =

(f) ∫e^(x + 1)^(1/2)/(x + 1)^(1/2) dx =




(2) Use Part 2 of the Fundamental Theorem of Calculus to write an antiderivative of f(x) = ln(cos(x) + 2)




(3) Find the following definite integrals. (You may use any technique. Sometimes area may be the best approach.)

(a)   ∫__ (-1)^1 (x + 1) dx =


(b)    ∫__1^31/x^2dx =


(c)  ∫_0^(1/2) 1/(1 - x^2)^(1/2) dx =


(d)  ∫__0^1 (x + 1)^(-1) dx =


(e)  ∫_ (-2)^( 2) (4 - x^2)^(1/2) dx =

(f)  ∫__ (-1)^1 | x | dx =


(4)  Use the definition of the definite integral to write  ∫_1^( 2) (ln(x) + x )^(1/2) dx   as a limit of Riemann sums.
(You do not need to find the value of this integral -- just write down the limit,) 

 


(5)  Find the average value of f(x) = x^(1/2) on the interval [0, 4].






(6) Find the derivatives of the following functions.

(a)   F(x) =    ∫__2^xtan (t^(1/2)) dt      


(b)   G(x) =    ∫__2^ln(x) tan(t^(1/2)) dt    


(7) Suppose a particle moves along the s-axis in such a way that its acceleration at time t seconds is a(t) = 4cos(t)units per second per second. Suppose that  v(0) = -1 and  s(0) = 3.  Find the position function s(t).

 

 



(8) Suppose that    ∫_3^2f(x) dx = 5  and  ∫_2^6f(x) dx = 7.

(a)    ∫_2^32f(x) dx =


(a)    ∫_3^6f(x) dx =