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The past exam paper of Calculus, key points are: Guarantee Accuracy, Integral, Right Endpoint, Approximate, Intervals, Necessary, Initial Value Problem, Evaluate, Trig Substitution, Conical Tank
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Math 106B - Final Exam, April 11, 2007
INSTRUCTIONS: Show all of your work and justify your solutions. Circle your final answers. Cross out any unnecessary work. The last page of this exam contains formulæ which may or may not be helpful in various problems.
0
e−x
2 dx. Using a right-endpoint sum to approximate I, how many intervals are necessary to guarantee accuracy to within ± 0 .005?
1 + y^2 x
, y(e) = 0.
dx (4 − x^2 )^3 /^2
. (Hint: Use a trig substitution.)
x.
(a) Find P 2 (x), the degree-2 Taylor polynomial of f (x) centered at x = 1.
(b) Use P 2 (x) to approximate
(c) According to Taylor’s Theorem, what is the maximum approximation error committed by P 2 (x) on the interval [1, 2]?
(d) Are your answers to (b) and (c) consistent? Explain.
∑^ ∞
k=
(−1)k k ln k
(a)
k=
k^2 + 4 100 k^2 + 3k + 5
(b)
k=
5 kk! (2k)!
(a) Evaluate lim x→ 0
x − x
3 3 −^ arctan^ x x^5
(b) Suppose g(x) = arctan(x). i. Evaluate g(99)(0).
ii. Evaluate g(100)(0).
Potentially useful formulæ
K 1 (b − a)^2 2 n
. • |I − Rn| ≤
K 1 (b − a)^2 2 n
K 2 (b − a)^3 12 n^2
. • |I − Mn| ≤
K 2 (b − a)^3 24 n^2
∫ (^) b
a
1 + (f ′(x))^2 dx
sinn(x)dx =
− sinn−^1 (x) cos(x) n
n − 1 n
sinn−^2 (x)dx, for n > 0.
cosn(x)dx =
cosn−^1 (x) sin(x) n
n − 1 n
cosn−^2 (x)dx, for n > 0.
tann(x)dx =
tann−^1 (x) n − 1
tann−^2 (x)dx, for n 6 = 1.
secn(x)dx = secn−^2 (x) tan(x) n − 1
n − 2 n − 1
secn−^2 (x)dx, for n 6 = 1.
sec(x)dx = ln | sec(x) + tan(x)| + C.
∑^ n
k=
f (k)(x 0 ) k!
(x − x 0 )k. • |f (x) − Pn(x)| ≤
Kn+ (n + 1)!
|x − x 0 |n+1.
k=
(−1)kx^2 k+ (2k + 1)! = x −
x^3 3!
x^5 5! −... on (−∞, ∞).
k=
(−1)kx^2 k (2k)!
x^2 2!
x^4 4!
−... on (−∞, ∞).
k=
xk x!
= 1 + x + x^2 2!
+... on (−∞, ∞).
k=
(−1)kx^2 k+ 2 k + 1
= x −
x^3 3
x^5 5
−... on [− 1 , 1].
