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For what values of x does the series \sum_{n=1}^{infty} (2x-1)^n/n converge?
Answer : 0<=x<1 | train | stemQ | Calculus II | [
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Find the second order Taylor polynomial of arctan(x) about the center a =1/2
Answer : arctan(1/2) + 4(x- 1/2)/5 -8*(x-1/2)^2/25 | train | stemQ | Calculus II | [
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Find the limit as x goes to 0 of (e^(2x) -cos(x) - sin(2x))/(ln(1+x) -x)
Answer : -5 | train | stemQ | Calculus II | [
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Find all real solutions to the differential equation cos^2(x)dy/dx + y = e^(tan(x))
Answer : e^(tan(x))/2 + Ce^(-tan(x)) | train | stemQ | Calculus II | [
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Find all real solutions to the differential equation d^2y/dx^2 + dy/dx - 2y = e^(3x)
Answer : C_1e^(x) + C_2e^(-2x) +e^(3x)/10 | train | stemQ | Calculus II | [
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Find the length of the curve given in parametric form by {x = 2(t^2 - 1)^(3/2), y=3t^2 where 2<=t<=3
Answer : 38 | train | stemQ | Calculus II | [
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Find the volume of the solid obtained by revolving the region under the curve y=cos(x) and above the x-axis, for 0<=x<=pi/3, about the line x=-1.
Answer : (pi^2)/(sqrt(3)) + pi*(sqrt(3)-1) | train | stemQ | Calculus II | [
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Evaluate the indefinite integral of sin(ln(t))dt
Answer : (t*sin(ln(t)) -t*cos(ln(t)))/2 +C | train | stemQ | Calculus II | [
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Evaluate the indefinite integral of dx/(sqrt(9x^2 + 16))
Answer : ln(3x + sqrt(16+9x^2))/3 + C | train | stemQ | Calculus II | [
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Approximate the following integral with an error less than 10^-3.
The definite integral of cos(sqrt(t))dt from 0 to 1/10.
Answer : 39/400 | train | stemQ | Calculus II | [
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Find the limit as x goes to 0 of (xcos(x) - xe^(-x^2))/(sin^3(x))
Answer : 1/2 | train | stemQ | Calculus II | [
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The region bounded by the curve y = x^3 + 1, the line x=0, and the lne y=9 is revolved around the line x=3. Find the volume.
Answer : 264pi/5 | train | stemQ | Calculus II | [
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Find all real solutions to the differential equation xdy/dx +2y = sin(x)
Answer : (-xcos(x) +sin(x) + C)/x^2 | train | stemQ | Calculus II | [
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Find all complex numbers z satisfying the equation (2z-1)^4 = -16. Express your answers in the form a+ib, where a and b are real.
Answer : 1/2 + sqrt(2)/2 + i*sqrt(2)/2
1/2 - sqrt(2)/2 + i*sqrt(2)/2
1/2 + sqrt(2)/2 - i*sqrt(2)/2
1/2 - sqrt(2)/2 - i*sqrt(2)/2
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Write AC or CC or D to indicate whether the given series is Absolutely Convergent or Conditionally convergent or Divergent.
\sum_{n=1}^{infty} (2^(2n) + (-5)^n)/5^n
Answer : D | train | stemQ | Calculus II | [
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Write AC or CC or D to indicate whether the given series is Absolutely Convergent or Conditionally convergent or Divergent.
\sum_{n=1}^{infty} (-1)^n(2^n)/(n!)^(1/3)
Answer : AC | train | stemQ | Calculus II | [
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Let C be the portion of the curve y=x^3 between the points (1,1) and (2,8). Find the area of the surface generated by rotating C about the x-axis.
Answer : pi*(sqrt(145)^3 - sqrt(10)^3)/27 | train | stemQ | Calculus II | [
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Find the limit as x goes to 0 of (e^(2x^2) - 1- 2x^2)*(cos(x)-1)/(x^4 * (sin(3x) - ln(1+3x)))
Answer : -2/9 | train | stemQ | Calculus II | [
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Estimate √9.2 using the linearization of the function √x at x = 9.
Answer : L(9.2)=3+0.2/6=3&1/30 =3.0333.... | train | stemQ | Calculus for Wharton Students | [
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“Dollar Bill” Stearn was convicted of tax evasion. As of January, 2017, he owed the U.S. Treasury a total of $14 million in back taxes. The payment agreement requires him to pay back $1.5 million each year on January 1, beginning in 2018. Federally mandated interest charges add 10% to his bill every December 31.
(a) Ho... | train | stemQ | Calculus for Wharton Students | [
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Let f(x,y) = x/y. Compute ∇f.
Answer : $$
\nabla f=\frac{1}{y} \hat{\mathbf{i}}-\frac{x}{y^{2}} \hat{\mathbf{j}}
$$ | train | stemQ | Calculus for Wharton Students | [
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An estuary contains 10 kg. of PCB (toxic waste) dispersed in a total of 3 MCF (million cubic feet) of mud. Influx of muddy water will add to the mud volume at the rate of 1 MCF / year. Superfund begins to clean up the site using an enzyme that breaks down (hence removes) the PCB at a rate proportional to the concentrat... | train | stemQ | Calculus for Wharton Students | [
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Use linearization to estimate log_10 1.069.
Answer : 0.03 | train | stemQ | Calculus for Wharton Students | [
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Give the general solution of the differential equation (1+t) y'+y=\sqrt{t}
Answer : $$
y(t)=\frac{2 t^{3 / 2}+C}{3(t+1)}
$$
where $C$ is any real number. We can also write this as
$$
y(t)=\frac{2}{3} \frac{t^{3 / 2}}{t+1}+\frac{c}{t+1}
$$
where $c$ is any real number.
| train | stemQ | Calculus for Wharton Students | [
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The price of a turkey is proportional to its weight and inversely proportional to the square of its age. Jack’s mother gives him enough money to buy a 10 pound turkey that is one year old. When Jack gets to the fair, he realizes that he needs a turkey that is slightly bigger. How much older will the turkey have to be p... | train | stemQ | Calculus for Wharton Students | [
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Compute the gradient of the function f (x, y) =x^2/ 8√y .
Answer : ∂f 2x x ∂x = 8√y=4√y
∂f x2
∂y = −16y3/2 .
xˆ x2ˆ Therefore the gradient of f is 4√y i − 16y3/2 j. | train | stemQ | Calculus for Wharton Students | [
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Compute the indefinite integral x−3 ln x dx.
Answer : This is a straightforward integration by parts. Let u = lnx and dv =
x−3dx so that v = −(1/2)x−2 dx and dv = x−1 dx. Then
x−3lnxdx=−1x−2lnx+1 x−2(x−1dx)=−1x−2lnx−1x−2. | train | stemQ | Calculus for Wharton Students | [
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Suppose x and y are related by ln x = 3 ln y. State in words the relation between x and y.
Answer : Exponentiate both sides with base $e: x=e^{3 \ln y}=\left(e^{\ln y}\right)^{3}=y^{3}$ In words, $x$ is equal to the cube of $y$, or $y$ is equal to the cube root of $x$ (either one of these is fine). | train | stemQ | Calculus for Wharton Students | [
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Find a function $g(x)=c x^{p}$ such that $f(x) \sim g(x)$ as $x \rightarrow \infty$ where
$$
f(x)=\frac{\sqrt{4 / x}}{\frac{3}{x}+\frac{5}{x^{2}}} .
$$
You do not have to prove your answer but we should be able to see how you got it.
Answer : $f(x) = \frac{2}{3} x^{1 / 2}$ | train | stemQ | Calculus for Wharton Students | [
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Use integrals to give upper and lower bounds for this sum.
99 √ mm.
m=16
These should be given as analytic expressions, with no integrals left un- evaluated, simplified when possible. Please include a sketch on the facing page (back of problem 6) or an extra piece of paper.
Answer : Upper bound:≈39,600
Lower bound:... | train | stemQ | Calculus for Wharton Students | [
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1... |
Let g(x) be the time in hours it takes to drill a one foot diameter shaft x meters deep into a bed of limestone. State an interpretation for the inverse function g−1.
Answer : g−1(t) is the length in meters of a shaft with diameter one foot that can be drilled into a bed of limestone in t hours. | train | stemQ | Calculus for Wharton Students | [
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1... |
Compute the indefinite integral of ln(1+x)*sqrt(1+x)
Answer : 2/3*(x + 1)^(3/2) * ln(x + 1) − 4/9*(x + 1)^(3/2) + c | train | stemQ | Calculus for Wharton Students | [
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Find c and p such that x^(-2)/(2+1/x)∼cx^p as x→∞.
Answer : c = 1/2 and p = −2 | train | stemQ | Calculus for Wharton Students | [
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] |
Compute the indefinite integral of x2e^(−x).
Answer : −x^2e^−x − 2xe^−x − 2e^−x + constant | train | stemQ | Calculus for Wharton Students | [
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Compute the quadratic (n = 2) Taylor polynomial P2(x) for the function x2/3 at the value a = 8.
Answer : Since f(x) = x2/3 has
′ 2x−1/3 ′′ 2x−4/3
f(x)= 3 andf (x)=−
P2(x) = f(8) + f′(8)(x − 8) + f′′(8)(x − 8)2
one computes
9 , =4+1(x−8)− 1 (x−8)2. | train | stemQ | Calculus for Wharton Students | [
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284,
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198,
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Write a differential equation for this scenario. You do not have to solve the differential equation but you must give the interpretation of all variables and constants, their units, and indicate which is the dependent and the independent variable.
The amount of an endowment increases due to investment at a rate proport... | train | stemQ | Calculus for Wharton Students | [
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A revenue forecast for a technological innovation forecasts earnings of n^(−2/5) million dollars in year n.
(a) Express the total possible earnings of this innovation (assuming you can continue marketing it forever) as an infinite series in Sigma-notation.
Answer : An observation tells us the series should be \Sigma^... | train | stemQ | Calculus for Wharton Students | [
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A dart lands uniformly somewhere in the triangle bounded by the x-axis, the y-axis and the line 5x + 2y = 10. Compute the probability density of the location of the dart.
Answer : Since the dart lands uniformly inside the shadow area, the density is just 1/Sshadow = 1/5 anywhere insider the shadow area, and zero el... | train | stemQ | Calculus for Wharton Students | [
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What is the (approximate) relation between x and y if ln y = 4.6 + 2 ln x?
Answer : y =~ 100x^2 | train | stemQ | Calculus for Wharton Students | [
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A retirement account growing at an exponential rate is expected to double in 14 years. Write an equation for the amount in the account after t years in terms of the starting contribution, assuming no further contributions. Be sure to give units for all constants.
Answer : α = ln 2/14 | train | stemQ | Calculus for Wharton Students | [
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Write an equation expressing the following scenario. Be sure to define variables and constants and to give units for each.
The number of troops necessary to patrol an area after combat units are with- drawn is proportional to the 1/2 power of the area being patrolled and inversely proportional to the length of time aft... | train | stemQ | Calculus for Wharton Students | [
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Compute $\lim _{x \rightarrow 0^{+}} x \ln x$
Answer : 0 | train | stemQ | Calculus for Wharton Students | [
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A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel (a solid disc) of mass M, radius R, anchored at its center but free to rotate. Which of energy, momentum and angular momentum is conserved for the bullet+wheel system? Give a few words of explanation.
Answer : Only angular mo... | train | stemQ | Fundamentals of Physics I | [
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A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel (a solid disc) of mass M, radius R, anchored at its center but free to rotate. Find ωf the final angular velocity of the wheel.
Answer : ωf = v/R * (1/(1+M/(2m)) | train | stemQ | Fundamentals of Physics I | [
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A block of mass M sits on frictionless table L meters from the edge. At t = 0 bullet of mass m and velocity v1 penetrates it from left and exits to the right with a speed v2. When will the block fly off the table? (Neglect loss of wood in block due to bullet penetrating it and the time it takes bullet to traverse bloc... | train | stemQ | Fundamentals of Physics I | [
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A block of mass M sits on frictionless table L meters from the edge. At t = 0 bullet of mass m and velocity v1 penetrates it from left and exits to the right with a speed v2. If the table has a height h how far from the edge of the table will it land? (Neglect loss of wood in block due to bullet penetrating it and th... | train | stemQ | Fundamentals of Physics I | [
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Consider the force F = 2xy^3*i + 3x^2y^2*j. Show that it is conservative.
Answer : ∂Fx/∂y = 6x*y^2 and ∂Fy/∂x = 6x*y^2, so F is conservative. | train | stemQ | Fundamentals of Physics I | [
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Consider the force F = 2xy^3*i + 3x^2y^2*j. What is the potential energy U(x, y) associated with it ?
Answer : U(x, y) = −x^2*y^3 | train | stemQ | Fundamentals of Physics I | [
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Consider the force F = 2xy^3*i + 3x^2y^2*j. What is the work done by the force along a path y = x^123456789 joining (0, 0) to (1, 1)?
Answer : 1 | train | stemQ | Fundamentals of Physics I | [
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Why can a body with total energy E < 0 not escape to infinity?
Answer : For a body bound in a gravitational potential, E = 1/2*m*v^2 −G*M*m/R. If R = ∞ then the potential energy term is zero and E will necessarily be greater than or equal to zero. Hence, for a mass with E < 0, R must always have a finite value and th... | train | stemQ | Fundamentals of Physics I | [
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A mass m tethered to a massless string is spinning in a vertical circle, keeping its total energy constant. Find the difference in the (magnitude of ) the tension between the top most and bottom most points.
Answer : 6m*g | train | stemQ | Fundamentals of Physics I | [
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The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time difference between the events according to an observer mov... | train | stemQ | Fundamentals of Physics I | [
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A muon has lifetime of 2*10^−6 s in its rest frame. It is created 100 km above the earth and moves towards it at a speed 2.97 * 108m/s. At what altitude does it decay? According to the muon, how far did it travel in its brief life?
Answer : The muon decays at 95.8 km above the ground. According to the muon, it has tr... | train | stemQ | Fundamentals of Physics I | [
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Two rockets of rest length L0 are approaching the earth from opposite directions at velocities ±c/2. How long does one of them appear to the other?
Answer : 3/5 * L0 | train | stemQ | Fundamentals of Physics I | [
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A body of rest mass m0 moving at speed v collides with and sticks to an identical body at rest. What is the mass and momentum of the final clump?
Answer : Momentum = γ*m0*v
Mass = m0 * sqrt(2*(1 + γ)) where γ =1/sqrt(1 − (v/c)^2) | train | stemQ | Fundamentals of Physics I | [
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A steel beam of mass M and length L is suspended at its midpoint by a cable and executes torsional oscillations. If two masses m are now attached to either end of the beam and this reduces the frequency by 10%, what is m/M?
Answer : 0.039 | train | stemQ | Fundamentals of Physics I | [
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Consider a particle attached to a spring executing a motion x = A sin(ωt + δ) with A = .32 m. At t = 0, it is at x = −.07m and a velocity −2m/s. The total energy is 5.6J. Find (i) δ, (ii) f the frequency, (iii) k and (iv) m
Answer : (i) δ = 2.92
(ii) f = 1.01 cycles/sec
(iii) k = 109 N/m
(iv) m = 2.7 kg | train | stemQ | Fundamentals of Physics I | [
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A mass m moving horizontally at velocity v0 on a frictionless table strikes a spring of force constant k. It compresses the spring and then bounces back with opposite velocity. Assuming no loss of energy anywhere find out (i) how long the mass is in contact with the spring and (ii) the maximum compression of the spring... | train | stemQ | Fundamentals of Physics I | [
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Write the equation for a wave moving along +x with a amplitude .4m, speed 6m/s and frequency 17Hz. If these are waves on a string with mass per unit length µ = .02kg/m, what is the u, the energy per unit length ? What is the power being fed into the vibrating string?
Answer : ψ(x,t) = (0.4m) * cos[(17*pi/3 m^-1)x – (... | train | stemQ | Fundamentals of Physics I | [
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How far part are the nodes on a string 80 cm long vibrating at 1600 Hz assuming a wave velocity of 320m/s
Answer : 8 nodes spaced every 10 cm | train | stemQ | Fundamentals of Physics I | [
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A vertical tube of radius 1 cm, open at the top to the atmosphere, contains 2 cm of oil (ρoil = 0.82ρwater) floating on 3 cm of water. What is the gauge pressure (i.e., pressure in excess of the atmospheric) at the bottom.
Answer : 460 Pa | train | stemQ | Fundamentals of Physics I | [
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Find the lift on an airplane wing of area 50 m^2 if the velocity of air at the upper and lower parts is 90 m/s and 70 m/s respectively. Assume ρair = 1.16 kg/m^3.
Answer : 9.3 * 10^4 N | train | stemQ | Fundamentals of Physics I | [
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A horizontal pipe of diameter d1 = 10 cm carrying water has a constriction of diameter d2 = 4 cm. If P1 = 105 Pa and P2 = 8 × 104 Pa, what is the flow rate?
Answer : 8.1*10^(−3) m^3/s. | train | stemQ | Fundamentals of Physics I | [
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How much heat is needed to convert 1 kg of ice at -10 °C to steam at 100 °C?
Answer : 3030 kJ | train | stemQ | Fundamentals of Physics I | [
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If 400g of ice at -2 °C is placed in 1 kg of water at 21 °C what is the end product when equilibrium is reached?
Answer : 1.258 kg of water and 0.142 kg of ice, all at 0°C | train | stemQ | Fundamentals of Physics I | [
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What is the increase in entropy if one gram of ice at O °C is melted and heated to 50 °C?
Answer : 0.461 cal/K | train | stemQ | Fundamentals of Physics I | [
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Find the change in entropy if 500 g of water at 80 °C is added to 300 g of water at 20 °C.
Answer : 3.20 cal/K | train | stemQ | Fundamentals of Physics I | [
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