Test 1 Review Sheet Hints—Math 122
Spring 2005, Prof. Frank
(1) One is a function and the other is a number. Explain.
(2) Yes. Two antiderivatives must only differ by a constant, and that is absorbed by C.
(3) False.
(4) (b) is the correct one. The value of the integral is 700.
(5) These are all u-substitutions.
(a) Try u= sin x.
(b) Try u= 1 + √x.
(c) Try u= ln(ln x). Compare that to trying u= ln x, which also works.
(6) (a) Draw something that intersects twice between x= 1 and x= 5.
(b) Draw something where both fand gare entirely above the x-axis.
(c) Draw fabove the x-axis and make gbe the zero function g(x) = 0.
(d) Draw fabove the x-axis and gbelow it.
(7) Yes, yes, and yes. The formula doesn’t change, you always subtract the smaller from
the larger.
(8) No. Consider the statement of the theorem of Pappus.
(9) False. There was a homework problem like this.
(10) (a) 2 ∗π∗3∗4 = 24π(b) 2 ∗π∗(3√2) ∗4 = 24√2π
(11) Your class notes contain this information about ln x.
(12) You’ll need to consider the first and second derivatives.
(13) One way is an area interpretation. Another uses the fact that ln xis increasing.
(14) I’d plot several representative points, perhaps letting x= 0,1, e, 36,64, on some graph
paper. You know what the graphs of the two functions look like, so just connect the
dots with nice smooth curves. It will be obvious that √x > ln xonce you have done
this. Take the derivatives of both functions and notice that the derivative of √xis
always bigger than the derivative of ln x. Think about why that implies that the
square root function is always bigger than the natural log function.
(15) Each rectangle with have a base of length 0.5; draw a picture of the rectangles to
figure out the heights. You should get 57
60 <ln 3 <67
60.
(16) You can show it is always increasing and you can show it passes the horizontal line
test.
(17) You know that (1,0) and (e, 1) are on the graph of lnx. Thus you know that (0,1)
and (1, e) are on the graph of f−1.
(18)
(19) False. Consider the theorem on page 377.
(20) True. Think about how you get the graph of an inverse function.
(21) True. Think about the domain and range of fas compared to the range and domain
of f−1.
