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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.
- True or false? Justify your answer with a reason or counterexample.
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
- Evaluate the limit.
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
- Find the antiderivative. Use power series in (c).
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
- Evaluate the integral. a)
∫ 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
- Show that
∫ (^) π/ 2 0 sin x sin x+cos x dx^ =^
π
- (hint: substitute^ u^ =^
π 2 −^ x)
- Determine whether the integral converges or diverges. If it converges, find the value.
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
- The electric potential due to a charged conducting sphere is V (r) = (^8) πq 0 a
∫ (^) 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.
- A tank with the indicated shape is full of water. The tank dimensions are height H m, length L m, and width W m, the water density is ρ kg/m^3 , and the acceleration due to gravity is g m/s^2. Find the work done in pumping the water to the top of the tank.
H
L
W
- Two identical ions repel each other with force F = − q
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
- Determine whether the series converges or diverges. Justify your answer.
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
- Express the repeating decimal as a rational number (i.e. a ratio of two integers).
a) 0. 11111111... b) 0. 1212121212... c) 0. 4999999999...
- Find the sum of the series. a)
∑∞ n=
2 n n! b)^
∑∞ n=
1 3 n^ c)^
∑∞ n=
n 3 n
- It is known that
∑∞ n=
1 n^2 =^
π^2
- Use this to evaluate^
∑∞ n=
1 (2n+1)^2.
- For each series, find a bound for |s − s 10 | using the estimates derived in class.
a)
∑∞ n=
1 n^2 b)^
∑∞ n=
(−1)n n^2
- Two students walk towards each other at 2 mi/hr starting from a separation of 20 miles. At the same time, a dog starts running back and forth between the students at 10 mi/hr. Let D be the total distance the dog has traveled when the students finally meet. Express D as an infinite series and find the sum of the series.
- Winning a game of ping-pong requires a lead of two points, i.e. if the final score is tied, you must score two consecutive points in order to win the game. Suppose your probability of scoring a point is p, where 0 < p < 1. If the final score is tied, find the probability you will eventually win the game. Evaluate for p = 12 , 14 , 34. Interpret.
- Start with the closed interval [0, 1]. Remove the open interval (^13 , 23 ). That leaves the two intervals [0, 13 ] and [^23 , 1]. Remove the middle third of those. That leaves four intervals. Remove the middle third of those. Continue the process indefinitely. The Cantor set is the set of all points remaining after all the intervals have been removed. (a) Show that the total length of all the intervals removed is 1. (b) Show that, nonetheless, the Cantor set contains infinitely many numbers.
power series, Taylor series
- Find the interval of convergence of the power series; find the function f (x) represented by the series; sketch the graph of f (x) and indicate the interval of convergence on the x-axis.
a)
∑∞ n=
xn^ b)
∑∞ n=
xn 2 n^ c)^
∑∞ n=
(x − 1)n^ d)
∑∞ n=
xn n e)^
∑∞ n=
nxn
- Find the power series representation for f (x) = (^1) −^1 x about x = 12.
- Find the Taylor series for sinh x and cosh x about x = 0.
- a) By squaring and adding the Taylor series for sin x and cos x, find the Taylor series for sin^2 x + cos^2 x, up to the O(x^6 ) term. b) Could you have predicted the answer to part (a)?
- Find the Taylor series for f (x) = e−x 2 about x = 0. Sketch f (x), T 0 (x), T 1 (x), T 2 (x) in a neighborhood of x = 0. Label each curve.
- Let f (x) =
{ 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).
- Find an approximate value for
10 which is accurate to within 0.005.
- Use the Taylor series for f (x) = ln(1 + x) about x = 0 to evaluate ln 32 to within 10−^3.
- Find the first two nonzero terms in the Taylor series for f (x) about x = 0.
a) e−x^ sin x b) (1 − cos x)/x c) tan x d) tan−^1 x
- The Bernoulli numbers Bn are defined by
x ex^ − 1
∑^ ∞
n=
Bn
xn n!
. Find B 0 , B 1 , B 2.
- Show that the following functions satisfy f (0) = 0, f ′(0) = 1. Find f ′′(0) in each case. If the functions are graphed in a neighborhood of x = 0, in what order do they appear (from top to bottom)? a) x b) sin x c) ln(1 + x) d) ex^ − 1
- Recall the Bessel function of order zero, J 0 (x) =
∑^ ∞
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.
- Let f (t) =
∑∞ 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?
- a) Show that
∫ (^) ∞
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))
- Use the 1st degree Taylor approximation for cos x about x = 0 to show that | cos π 5 − 1 | ≤ 12 (π 5 )^2. Derive a more accurate result using the 3rd degree Taylor approximation.
- Recall the error function, erf(x) =
π
∫ (^) x
0
e−t
2 dt. Find the first three terms in the Taylor
series for erf(x) about x = 0.
- In both cases below find the first three nonzero terms.
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
- Show that f (x + h) = f (x) + f ′(x)h + f^
′′(x) 2 h
(^2) + · · · (an alternative form of the Taylor series).
- The equation
x^2 (1 + )^2
- y^2 = 1 defines an ellipse in the xy-plane (assume 0 ≤ < 1).
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.
- The gravitational potential induced by two point masses m 1 , m 2 located at x 1 , x 2 on the x-axis is V (x) = − (^) |xGm−x^11 | − (^) |xGm−x^22 | , where G is the gravitational constant. For x → ∞, we have
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.)
