Calculus II (Math 242) - Test 1 (No Work - No Credit)

Problem 1. [10 Points] Find the partial sum decomposition of

Problem 2. [6 Points] For :r > 0 we defined the natural logarithm as:

j dtI

lnx= - . . 1 t

State what it means that a function is 1-1, and show that In.:r: has this property.

Problem 3. [18 Points] Calculate the followjng limits:

In:r (2) lim :re:J: and (:3) lim (1 + sin(4:r:))5/"'.(1) Cl~1Jo .yx .T--1--00 x-->o-

Problem 4. [8 Points] Graph and differentiate the function arctan(x:). Im­portant: Sho'vV your work!

Problem 5. [56 Points] Work out the follovving integrals:

3 3(1) Ie ." cos(5x)d:r (2) I sin JX d:r (3) I sec":t dx (4) I tan x: &1.'

(5) j' /:1.'2 + 9 dx (6) j' ~ dx: (.7) j' .X -I 1 _ d:r:. x 4 1 + ijX x:2 + 4x + b

Problem 6. [14 Points] A home buyer can afford to spend no more than .$800.00 per month on mortage payments. Suppose the interest rate is 9% and t be term of the mort.gage is 20 years. Assume that the interest is paid continuously and that payments are also made continuously.

1. Express the situation as an init.ial value problem.

2. Determine t.he ma.,'(imum amount. this buyer can afford to borrow.

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