1
warmup
Series of numbers
Real & complex analysis
Write the first three nonzero terms of the Maclaurin series for \(\sin^{-1} x\).
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2
core
The definite integral
Calculus
If $g$ is continuous, which of the following integrals are equal?
(a) $\int_{a}^{b} g(x) dx$
(b) $\int_{a+1}^{b+1} g(x - 1) dx$
(c) $\int_{0}^{b-a} g(x + a) dx$
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3
core
Differential equations
Calculus
Solve the boundary value problem
\[ y^{\prime\prime} - 7y^{\prime} + 12y = 4e^{2x}, \quad y(0) = 3, \quad y(1) = 5e^2. \]
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4
core
Limits
Calculus
Find $\lim_{x\to+\infty} \frac{2x+5}{x^2-7x+3}$.
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5
core
Multiple integrals
Calculus
\[\begin{aligned} &Let a transformation T be defined as T(u, v) = (x, y) where x = u + v, y = 3v. \\ &Find the image of the
rectangle G = {(u, v): 0 \le u \le 1, 0 \le v \le 2} from the uv-plane after the transformation into a region R in
the xy-plane. \\ &Show that T is a one-to-one transformation and find T^{-1}(x, y).\end{aligned}\]
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6
warmup
Multivariable calculus
Calculus
Calculate the work done by field \( \mathbf{F}(x, y) = \langle 2x, 3y \rangle \) on a particle that traverses the unit circle. Assume the particle begins its movement at \( (-1, 0) \).
calculus-iiivector-calculusline-integralworkconservative-vector-fieldclosed-loopfundamental-theorem-of-line-integralsunit-circleparameterizationdot-productpartial-derivativesevaluate-integral
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Work Done by a Conservative Field on a Unit Circle →
7
warmup
Parametric equations and vector functions
Calculus
Solve \( x - y^2 \frac{dy}{dx} = 0 \).
differential-equationsseparable-differential-equationsintegrationpower-rulefirst-order-odecalculus-icalculus-iiseparation-of-variablesgeneral-solutionarbitrary-constantalgebraic-manipulationimplicit-solution
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Solving a Simple Separable Differential Equation →
8
warmup
Logarithmic and exponential functions
Calculus
If a and b are positive, find the arc length of \[\begin{aligned}y &= \frac{x}{b} \\ &= \frac{a}{8} (\frac{x}{a})^2 - \frac{1}{8} \frac{a^2}{b^2} \ln (\frac{x}{a})\end{aligned}\] from x=a to x = 3a.
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9
warmup
Trigonometric and inverse trigonometric functions
Calculus
Calculate $\lim_{x\to 0} \frac{1 - \cos x}{x}$.
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10
hard
Vector spaces
Linear algebra
[T] \( r = 2 \sec \theta \)
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