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Unformatted text preview: Solomons SHORT CALCULUS
MAT 16B, Section 001 — Fall 2011
Midterm 1 DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO NAME: ...................................................... ..
ID NUMBER: ...................................................... ..
SIGNATURE: ...................................................... ..
Keep you student’s :i.d. Visible in front of you.
No books, no calculators or electronic devices allowed.
Turn your cell phone off. No communication among the students will be tolerated.
SHOW ALL YOUR WORK Problem Total Points Student’s Score Total 42 (+4) PROBLEM 1. [10 pts]
i.[4 pts] True or false? Circle the right answer: 111(x2) _ Inst 2
G — (G ) e(r+l)lnace—zlna: : m +1 True ln((21: — = ln 2x2 — ln3117 True
1 1n x/err = 5(ln(zr)+1) ( True) ii.[2pt] Expand the logarithmic expression as much as possible: ln (1U («x i 411 @034) iii.[2pt] Write the following expression as the logarithm of a single quantity: 1
2 (1112? + E); ln(2:2: — 3)) (M is)?” iv.[2pt] Solve for s: 365.1; 9} =p SZ= PROBLEM 2. [6pts]
i.[2pts] Find the slope of the tangent line to the function g(:r) = ate2$2 at the point (1, 1/e2) ii.[4pts] Use implicit differentiation to ﬁnd y’ in the domain y > 0,1L‘ > 0: e33 + 21n(272y) = :1: + y2 PROBLEM 3. [5pts]
[3pts] You have $1000 at your disposal, and you have four different options: i. A ﬁrst type of bank account with an interest rate of 3% per year, compounded once a
year, ii. A second type of bank account with an interest rate of 3% per year, compounded
monthly (12 times per year), iii. A third type of bank account with an interest rate of 2% per year, compounded contin
uously. In each situation, give a formula for the amount of money S you have after 5 years (You do
not need to evaluate the numerical expressions) S = (000 (140.03.)? v loo (a u 40,?) 0019 0.1
Siii: [0008. b (0008 [2pts] Compare options i and ii and write a short paragraph to explain why one grows faster
than the other. ' SE, >§; KI?) COWVOMCLiaé "Hg (463d?) imi‘UEJl (More ﬁfe maﬁa ( Vat 05,
{LL interest 4‘“; (gear él/Céaa Values Evieresl ’Xor {Lg W PROBLEM 4. [6pts]
After 10 years, 60% of the initial amount of a radioactive substance remains. What is the
half—life of the substance? You do not need to numerically evaluate the logarithms. Hint: Recall that the half—life is the time after which half of the initial amount has decayed Tlolet "gill: la mm ol mm W 4*“ Wm /'
So
I: C0 :3” C_=I (
‘ (oh a i _ .051;
0(3‘ 8 D V< ‘ (0
wk
Hamel e l PROBLEM 5. [10 pts]
Compute the following antiderivatives /15‘r:m+4dm= RGXHL ‘ Xs+ Ell/14')“ +C. /:v2—r—5_Lv3+6dx: szg‘gdh Ski» § ~{xk2x+ C.
dm / (5eigej2>sd$= 1 (fuck * gay. ~U‘L+C 75%“)
a: u: EekL ', if; : (Fey 3 ﬁmx ~ iéguy‘m =§KMH
as (A: ’Lx‘JrH " i: “X /x<hias>2d“”= Xi; mitt“ : “31%” : _b;l>(+ C
a‘ “tax a it §3L< 11 PROBLEM 6. [5pts]
i.[3pts] Find the function f(:c) such that f’(m) = 31:2 — 1 and the value of f is 2 when a: = 1. ﬁg) :5 at, Aukehzivahw oﬁ 3xLy 'ﬂhayt)ar¢
ﬁx): XgX + C
at >r=(: 4m: 1“ C Mae 1 ﬂ") = Xg"x+z' ii.[2pts] True or false? Circle the right answer and justify your choice below: /$2exda:=(2—2x+x2)ex—+—C @ False
we M‘ i QQL—LX'bXVkET‘IL C)
{gun} +Lz~1m~ﬂex : x‘ex
Wat 15 4k ‘(u am! wech 13 PROBLEM 7. [4pts, extra credit]
i.[2pts] Let = fe“’”2da:. Compute f”(3:) 3% who“. Mime”
a4 tempt: w s 4%“ V ii.[2pts] Let be a given function. Suppose F and C(x) are two functions such that
F’(x) = and G’(LE) = Compute — FPO an“; COO [961,3 4% «Avidcribl'ibwrjx “VAR?
(My Li?) a cousbﬂr ' :FGQ‘ a C 15 ...
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