Key to Practice Midterm I

Problem 1

A.  True.  For monotonically increasing, if a < b, then f(a) < f(b) so if f(a) = f(b) then b is not greater than a.  Since f is monotonically increasing, b = a.

B.  False, for example, ln (1 + 1) = ln(2), but ln(1)ln(1) = 0

C.  True, the curves intersect at (0,1) and f'(0) = 1 and g'(0) = -1 are negative reciprocals.

Problem 2

A.  (lnx)^x(ln(lnx) + 1/lnx)    B.  2x     C.  3^xln3 - 3x² + x^x(1 + lnx) 

Problem 3

A.  2/5 (x + 1)^5/2 - 2/3 (x + 1)^3/2 + C  

B.  1/2 e^2x-1 + C 

C.  -ln|1 - e^x| + C

Problem 4

A.  dy/dt = kt^1/2y, y(0) = 2, y(9) = 50, y represents the number of words the child knows at time t (months since first birthday).

B.  2e^{.12t^1.5}  

C.  approximately 284 words.

Problem 5

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