Calculus II
Final Exam
May 23, 2002
Name____________________
R.  Hammack
Score______


(1) Evaluate the following integrals.
(a)   ∫ (1/x^2 + 1/x + x^2 + 5) dx =


(b) ∫ (ln x)^(1/2)/x dx =




(c)   ∫ x/(1 + x^2) dx =


(d)   ∫ 1/(1 + x^2) dx =



(e)   ∫ x^4 e^x^5 dx =




(f)    ∫ e^(5 x) dx =


(g)   ∫ x e^(5 x) dx =



(2) Evaluate the following definite integrals

(a)  ∫ _ (-1)^2 x^3 dx =


(b) ∫ _ (-1)^2 | x^3 | dx =


(c) ∫ _ 0^π^(1/2) 5 x cos(x^2) dx =


(3) Evaluate the following integrals.
(a) ∫ sin^3(x) cos^2(x) dx =

 



(b) ∫ dx/(x^2 + 3 x - 4) =




(4) Evaluate the following definite integrals. (Notice that they are both improper.)
(a)   ∫ _ 1^∞ e^(-x) dx =




(b) ∫ _ 0^(π/6) (cos x)/(1 - 2 sin x)^(1/2) dx =

 

 

(5) Consider the region between the graph of y = x - x^2 and the x-axis.


(a) This region is rotated around the y-axis. What is the volume of the resulting solid?





(b) This region is rotated around the x-axis. What is the volume of the resulting solid?

 

 

(6) Find the area enclosed between the curves y = x^2,   y = x^(1/2),   x = 1/4,  and  x = 1.

 



(7) Find the derivative of the function   F(x) = ∫ _ 2^x^3 (1 + sin t)^(1/2) dt

 

 

(8) Decide if the following sequences converge or diverge. In the case of convergence, find the limit.

(a) {(8 e^n)/(e^n + 1)} _ (n = 1)^∞


(b)  {1/n ln(1/n)} _ (n = 1)^∞

(9) Decide if the following series converge or diverge.
(a) Underoverscript[∑ , k = 1, arg3] (3/5)^k (k - 1)

(b)  Underoverscript[∑ , k = 1, arg3] 1/k^(1/2)


(c)  Underoverscript[∑ , k = 1, arg3] (-1)^k/k^(1/2)


(d)  Underoverscript[∑ , k = 1, arg3] 3/(2^k + k^(1/2))

 


(10)   Both of the following series converge. Say what number they converge to.

(a)   3 - 3/2 + 3/4 - 3/8 + 3/16 - 3/32 + 3/64 - ...  ]


(b)   1 + 1/1 ! + 1/2 ! + 1/3 ! + 1/4 ! + 1/5 ! + 1/6 ! + 1/7 ! + ...   

 


(11) The remaining problems on this exam are based on the function f(x) = ln(1 + x).

(a) Derive the Maclaurin series for the function f(x) = ln(1 + x).   Show all of  your work.

 





(b) Find the interval of convergence of the Maclaurin series from part (a).  Show all of  your work.

Let's use the ratio test to check for absolute convergence.

 



(c) Use your answer to part (a) above to express ln(2) as an infinite series.

ln(2) =



(d) Use your answer to part (a) above to find a power series representation of the function ln(1 + x^2).

ln(1 + x^2)=



(e) Write ∫ _ 0^1 ln(1 + x^2) dx  as an infinite series. (You may just write out the first 5 or 6 terms.)