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MATH 141: FINAL EXAM
Name
Instructions and Point Values: Put your name in the space provided ab ove. Check that your test contains 14 di erent pages including one blank page. Work each problem b elow and show ALL of your work. Unless stated otherwise, you do not need to simplify your answers. You should NOT use Calculus material in your answers that are unrelated to this course. Do NOT use a calculator.
There are 300 total p oints p ossible on this exam. There are 2 parts. The rst part consists of 20 problems each worth 10 p oints. The second part consists of 5 problems each worth 20 p oints.
PART I. Calculate each of the following. An asterisk () next to a problem numb er indicates that you do not need to show work for that problem. You should show work for every other problem.
(1) The equation of the tangent line to the graph of y = x^2 at the p oint (1; 1).
(2) lim t! 2
t^2 4 t^2 + 4
(3) lim x! 0
jxj x
(4) lim x! 1 +
x^2 1 p x 1
(5) lim x! 0
x^2 (cos x) 1
(9)^
d dx
p
x (x + 1)^20
(10) f 0 (t) if f (t) = sin(3t + 1)
(11) y = f (x) if xy 0 + xy 2 =
y 2 x
and if f (1) = 1
d^5 dx^5
2 x^4 + 33 x^3 23 x + 7
d dx
(f (1=x)) if f 0 (x) = 10 x
Z = 4
0
sin(4t) dt
d dx
Z x^2
x
cos
p
t dt
Z 2
0
d dt
(cos
p
t)
dt
(20) The area of the region b ounded by the graphs of x = y 2 and x = 1.
PART I I. Answer each of the following. Make sure your work is clear. If you do not know how to answer a problem, tell me what you know that you think is relevant to the problem. If you end up with an answer that you think is incorrect, tell me this as well. Better yet, tell me why you think it is incorrect. In other words, let me know what you know.
(1) The p oint (1; 1) is on the graph of x^2 y + xy 3 = 2. Calculate the equation of the tangent line to the graph of x^2 y + xy 3 = 2 at the p oint (1; 1).
(d) Find all the in ection p oints for y = f (x).
(e) On what intervals is the graph of y = f (x) concave up?
(f ) Graph y = f (x).
(3) Part (a) b elow can b e done with or without Calculus. If you happ en to know a formula for the distance describ ed in (a), you should justify that the formula works b efore using it. Part (b) b elow dep ends on Part (a), but you do not need to do Part (a) correctly to do Part (b) correctly. You should try Part (b) even if you feel you cannot do Part (a).
(a) Let P b e a p oint (a; b), and supp ose that P is to the right of the line y = x. This means a > b. Show that the distance (in other words, the \minimal" distance) from P to the line y = x is (a b)=
p
(b) Consider the p oints (a; b) on the graph of y = x^3 with 0 a 1. Determine which of these p oints (a; b) is the greatest distance from the line y = x.
(5) The graph of y =
sin x x
intersects the x-axis at x = 2 and x = 3 (and at some other
p oints to o). The graph is ab ove the x-axis for 2 < x < 3 . Let R b e the region b elow
the graph of y =
sin x x
and ab ove the x-axis (you do not need to know exactly what this
region lo oks like). Calculate the volume of the solid obtained by revolving the region R ab out the y -axis. Simplify your answer.
