1
warmup
Sequences
Real & complex analysis
Study the series $\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\dots+\frac{1}{(2n-1)(2n+1)}+\dots$
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2
core
Antiderivatives
Calculus
Find $\int (\sqrt{x} + 1)^2 dx$.
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3
core
Differential equations
Calculus
Find the Laplace transform of
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4
core
Multiple integrals
Calculus
Evaluate the integral \(\iint_{R} (4-x^2-y^2)dA\) where \(R\) is the circle of radius 2 on the xy-plane.
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5
core
Multivariable calculus
Calculus
Give the \"upward\" orientation of the graph of $f(x, y) = xy$.
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6
core
Parametric equations and vector functions
Calculus
Show that the total length of the ellipse \(x = 4 \sin \theta, y = 3 \cos \theta\) is \(L = 16 \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \theta} d\theta, \text{ where } e = \frac{c}{a} \text{ and } c = \sqrt{a^2 - b^2}\).
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7
core
Parametric equations and vector functions
Calculus
\(x = 2a \cos t - a \sin(2t), y = b \sin t, 0 \le t < 2\pi\) [T] (the "teardrop")
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8
core
Parametric equations and vector functions
Calculus
Find the area of the surface generated by revolving \(x = t^2, y = 2t, 0 \le t \le 4\) about the x-axis.
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9
core
Recurrences
Discrete mathematics
Solve the equation $Df = 3f$ if $f(0) = 2$.
differential-equationsinitial-value-problemexponential-functionfirst-order-linearcalculus-iderivativeschain-rulefind-function
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Solving a Simple First-Order Differential Equation →
10
warmup
Vector spaces
Linear algebra
Assume the quarterback and the receiver are in the same place as in the previous example. This time, however, the quarterback throws the ball at velocity of 40 mph and an angle of 45 degrees. Write the initial velocity vector of the ball, \( \mathbf{v} \), in component form.
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