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Applied Honors Calculus II. Final Exam Review Sheet. Fall 2022. For full credit, justify your answer, and give the units if appropriate. 1. True or false?
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Math 156 Applied Honors Calculus II Final Exam Review Sheet Fall 2022
For full credit, justify your answer, and give the units if appropriate.
a) (^1) n + (^) n^2 + (^) n^3 + · · · + nn = n+1 2
b) If ∆x = b−na and xi = a + i∆x, then lim n→∞
∑^ n i=
f ′(xi)∆x = f (b) − f (a).
c) If the integral
∫ (^) b a f^ (x)^ dx^ is approximated by the right-hand Riemann sum and the number of intervals n is doubled, then the error decreases by approximately 14.
d) If f (0) = f (1) = g(0) = g(1) = 0, then
∫ (^1) 0 f^ (x)g
′′(x)dx = ∫^1 0 f^
′′(x)g(x)dx.
e)
∫ (^) ∞ 0 dx x^2 is a convergent improper integral. f) A spring has natural length 20 cm. If 2 Joules of work are needed to stretch the spring from length 20 cm to 30 cm, then 4 Joules of work are needed to stretch it from length 30 cm to 40 cm.
g) The center of mass of the region {(x, y) : − 1 ≤ x ≤ 1 , 0 ≤ y ≤ cosh x} is (x, y) = (0, 12 ).
h) If f (x) is the pdf of a random variable with mean μ, then f (x) has its maximum value at x = μ.
i) If f (x) is the pdf of a random variable with mean μ, then
∫ (^) ∞ −∞(x^ −^ μ)f^ (x)dx^ = 0. j) If a radioactive material has a half-life of 100 years and a sample has mass 1 kg, then there will be 0.25 kg remaining after 400 years.
k) If $1000 is invested at 5% interest compounded continuously, then after 2 years the investment is worth between $1105 and $1112.
l) y(t) = 0 is a stable constant solution of the differential equation y′^ = y(1 − y^2 ).
m) If a differential equation y′^ = f (y) has a unique constant solution y 1 (t) = c, and y 2 (t) is any other solution with initial condition y 2 (0) 6 = c, then lim t→∞ y 2 (t) = c.
n) If a differential equation y′^ = f (y) is solved by Euler’s method, and the step size ∆t decreases by a factor of 12 , then the error in the numerical solution increases by a factor of approximately 12.
o) If lim n→∞ an = 0 and lim n→∞ bn = ∞, then lim n→∞ anbn = 0.
p) If 0 ≤ an ≤ bn and
∑∞ n=
an converges, then
∑∞ n=
bn also converges.
q)
∑∞ n=
1 n^2 <^
∫ (^) ∞ 1
dx x^2 r) 1^ −^
1 2 +^
1 3 −^
1 4 +^ · · ·^ = 0
s)
∑∞ n=
1 n(n+1) =^
∑∞ n− 1
( (^1) n − (^) n+1^1 ) =
∑∞ n=
1 n −^
∑∞ n=
1 n+1 =^ ∞ − ∞^ = 0
t) The alternating series test can be used to show that
∑∞ n=
(−1)n^ diverges.
u) The ratio test can be used to show that
∑∞ n=
1 n^2 converges.
v) If the power series
∑∞ n=
cnxn^ converges for x = 1, then it also converges for x = −1.
w1) If the power series
∑∞ n=
cn(x − 1)n^ converges for x = 2, then it also converges for x = 12.
w2) The series
∑∞ n=
2 n(x − 1)n^ converges for 12 ≤ x ≤ 32.
x) (^) (1+^1 x) 2 =
∑∞ n=
(−1)n(n + 1)xn^ for |x| < 1 y) If f (x) = e−x
2 , then f (3)(0) = 0, f (6)(0) = −6.
z) 2 < e < 3 aa)
∫ (^1) 0 e
−x^2 dx > 2 3 bb)^
π 2 −^
1 3! (
π 2 )
π 2 )
π 2 )
cc) cosh^2 x − sinh^2 x = 1 dd)
∫ tanh x dx = sech^2 x
ee) If T 1 (x) is the first degree Taylor polynomial for f (x) at x = a, then the graphs of f (x) and T 1 (x) have the same slope at x = a.
ff) If
5 is approximated by T 1 (5), where T 1 (x) is the Taylor polynomial of degree one for
x at a = 4, then the approximation is larger than the exact value.
gg) 0. 895 < e−^0.^1 < 0. 905 hh)
1 + x^2 = 1 + x^2 + · · · ii)
∫ (^) π/ 2 0 sin (^2) θdθ = ∫^ π/^2 0 cos (^2) θdθ
jj) eπi/^4 + e−πi/^4 =
2 kk) cosh ix = cos x ll) log(−1) = πi mm) (−^12 + i
√ 3 2 )
nn) There are 56 ways of choosing 3 objects from a set of 8 objects, disregarding the order in which they are chosen.
oo)
( 6 3
) = 2 pp)
( 10 2
( 10 8
) qq)
( 7 3
)
( 7 4
( 8 4
) rr)
∑^10 n=
( 10 n
) = 1024 ss)
∑k n=
(k n
) (−1)n^ = 0
a) 1 + 20222023 + (^20222023 )^2 + (^20222023 )^3 + · · · b) limn→∞
∑^ n i=
(1 + (^) ni ) · (^) n^1 c) limn→∞
∑^ n i=
1 1+ (^) ni^ ·^
1 n
d) lim x→ 0
sin x x e) lim x→ 0
1 −cos x x^2 f) lim n→∞(1 +^
x n )
2 n (^) g) lim x→ 0
√1+x− 1 x h) lim h→ 0
(x+h)^4 −x^4 h
i) lim h→ 0
f (x+h)−f (x) h j) lim h→ 0
f (x+h)− 2 f (x)+f (x−h) h^2 k) lim h→ 0
1 h
∫ (^) h 0 f^ (x)^ dx^ l) lim h→ 0
1 h^2
∫ (^) h 0 xf^ (x)^ dx
integration
a)
∫ e−xdx b)
∫ xe−xdx c)
∫ e−x 2 dx d)
∫ xe−x 2 dx e)
∫ x sin x dx f)
∫ e−x^ sin x dx
g)
∫ (^) dx 4 x^2 h)^
∫ (^) x 4+x^2 dx^ i)^
∫ (^) dx 4+x^2 j)^
∫ (^) dx √4+x 2 k) ∫ (^) dx 4 −x^2 l)^
∫ (^) dx 4 x−x^2 m)^
∫ (^) dx √ 4 x−x 2
n)
∫ sin^2 x dx o)
∫ sin^3 x dx p)
∫ sin^4 x dx
∫ 2 √ 3 0 √x^3 16 −x^2 dx^ b)^
∫ (^) ∞ −∞ x (^2) √^1 2 π e
−x^2 (^2) dx c)
∫ (^) ∞ −∞(x^ −^ 1) 2 1 2 √ 2 π e
−(x−1)^2 (^8) dx
∫ (^) π/ 2 0 sin x sin x+cos x dx^ =^
π
π 2 −^ x)
a)
∫ (^) ∞ 1 dx x^2 b)^
∫ (^) ∞ 1 dx x c)^
∫ (^) ∞ 1 dx x− 1 d)^
∫ (^1) 0 dx x^2 e)^
∫ (^1) 0 √dx x f)^
∫ (^1) − 1 dx x g)^
∫ (^) ∞ 0 x^2 dx (1+x^2 )^7 /^2
∫ (^) a −a dx (r^2 − 2 rx+a^2 )^1 /^2 , where q is the total charge on the sphere, a is the radius of the sphere, and r is the distance from the center of the sphere to a point in space. a) Evaluate V (r). Consider two cases, 0 ≤ r ≤ a and r > a.
b) Sketch the graph of V (r) for r ≥ 0.
2 4 π 0 r^2 , where^ q^ is the ion charge,^ r^ is the distance between the ions, and 0 is the free-space permittivity. The negative sign indicates a repulsive force. (a) An ion is held fixed at x = 0. Find the work done in moving another ion from x = 3 to x = 2. (b) An ion is held fixed at x = 1. Find the work done in moving another ion from x = 3 to x = 2. (c) Two ions are held fixed at x = 0 and x = 1. Find the work done in
steps with ∆t = (^1) n. Find the expression for un and evaluate the limit lim ∆t→ 0 un.
series
a)
∑∞ n=
1 2 n b)^
∑∞ n=
1 2 n^ c)^
∑∞ n=
1 n^2 d)^
∑∞ n=
(−1)n n^2 e)^
∑∞ n=
(−2)n n^2
a) 0. 11111111... b) 0. 1212121212... c) 0. 4999999999...
∑∞ n=
2 n n! b)^
∑∞ n=
1 3 n^ c)^
∑∞ n=
n 3 n
∑∞ n=
1 n^2 =^
π^2
∑∞ n=
1 (2n+1)^2.
a)
∑∞ n=
1 n^2 b)^
∑∞ n=
(−1)n n^2
power series, Taylor series
a)
∑∞ n=
xn^ b)
∑∞ n=
xn 2 n^ c)^
∑∞ n=
(x − 1)n^ d)
∑∞ n=
xn n e)^
∑∞ n=
nxn
{ e−^1 /x^ for x > 0 , 0 for x ≤ 0.
Evaluate the following limits.
a) limx→∞ f (x) b) lim x→ 0 +
f (x) c) lim x→ 0 +
f ′(x) d) lim x→ 0 +
f ′′(x) e) Sketch the graph of f (x).
10 which is accurate to within 0.005.
a) e−x^ sin x b) (1 − cos x)/x c) tan x d) tan−^1 x
x ex^ − 1
∑^ ∞
n=
Bn
xn n!
. Find B 0 , B 1 , B 2.
∑^ ∞
n=
(−1)nx^2 n 22 n(n!)^2
a) Evaluate
∫^1 0
J 0 (x) dx using 2 terms in the series. Find an upper bound for the error.
b) Show that J 0 (x) satisfies the differential equation xy′′^ + y′^ + xy = 0.
∑∞ n=
tn.
a) Show that f (t) satisfies the differential equation y′^ = y^2 with initial condition y(0) = 1.
b) Solve the differential equation for f (t) by separation of variables. Do you recognize the result?
∫ (^) ∞
0
sin x x
dx converges, but
∫ (^) ∞
0
∣∣ ∣∣^ sin^ x x
∣∣ ∣∣ dx diverges.
(hint: sketch the graph, express the integral as a series, bound the terms in the series)
b) Find the value of
∫ (^) ∞ 0
sin x x dx. (hint: let^ f^ (a) =^
∫ (^) ∞ 0
sin x x e
−axdx for a ≥ 0, evaluate f ′(a), then
find f (a), and finally evaluate f (0))
π
∫ (^) x
0
e−t
2 dt. Find the first three terms in the Taylor
series for erf(x) about x = 0.
a) expand
a a + b
in powers of
a b
assuming 0 < a < b
b) expand
R^2 − r^2 R
in powers of
r R
assuming 0 < r < R
′′(x) 2 h
(^2) + · · · (an alternative form of the Taylor series).
x^2 (1 + )^2
a) Find the intercepts on the x-axis and y-axis. Sketch the ellipse.
b) Let A() be the area of the ellipse. Express A() as a definite integral.
c) Find the first 2 nonzero terms in the power series expansion of A() about = 0.
V (x) ≈ ax + (^) xb 2 + (^) xc 3 + · · ·, where a, b, c,... are constants that depend on m 1 , m 2 , x 1 , x 2. Find a, b, c. (hint: set y = 1/x and expand V (x) in powers of y.)