MATHEMATICS H161--AU03    MIDTERM TWO

 

NAME _____________________________

 

 

Do all problems.  Show all steps for your work.  (Even if you get an answer just from a calculator, you need to explain why it is correct.)  Formulas for your use are on the back of the last page.

 

1.  Calculate the following integrals.

 

            (a) (5 points)  indefinite integral.  The integrand is  (tan3 x)/(sin2 x)

 

 

 

 

 

 

 

 

            (b) (5 points)  definite integral from  0  to pi/4  of the integrand in problem 1,

 

 

 

 

 

 

 

 

            (c) (5 points)  indefinite integral. The integrand is  1/(9x2 + 4)

 

 

 

 

 

 

 

 


2.  (5 points)  Evaluate the definite integral from  2  to  7  of the integrand

 

square root of (- x2 + 4x + 21)

 

 

 

 

 

 

 

 

 

 

3. (10 points)  Find the area between the curve  y = cosh x  and the x-axis  from  x = 0  to  x = 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

4. (10 points)  Find the length of the curve     y3/2       from  x = 0  to  x = 1.


5. (10 points)  Use the approximate value  ln 2 = 0.7 (not the value you would get from your calculator) to find the area between the graph of  y = 1/x  and the  x-axis from  x = 2  to  x = 8.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6. (10 points)  At  2:00 pm a count of a bacteria culture was  50 per square centimeter.  At

3:00 pm the count was 70 per square centimeter.  If the culture grows at the same rate until midnight of the same day, at what time (to the nearest 10 minutes) will the count be  300  per square centimeter?


7. (a) (5 points)  Evaluate the definite integral from  3  to  4  of the integrand  1/(x 2).

 

 

 

 

 

 

 

 

 

(b) (5 points)  Evaluate the definite integral from  - 4  to  - 3  of the integrand  1/(x 2).

 

 

 

 

 

 

 

 

 

 

 

 

8. (10 points)  State carefully the fundamental theorem of calculus.


9. (extra credit 10 points, but you can't get more than 100% = 85 points on the exam)  Prove the fundamental theorem of calculus.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

           

                                                     

 

 

 


ln 3 = 1.09861

 

ln 5 = 1.60944

 

ln 6 = 1.79176

 

ln 7 = 1.94591

 

sin2 x = (1 cos 2x)/2

 

cos2 x = (1 + cos 2x)/2

 

sin (x + y) = sin x cos y  +  cos x sin y

 

cos (x + y) = cos x cos y  -  sin x sin y