Topic 1: Introduction to Differential Equations (Separable and First-Order ODEs)
Easy Questions 🌱
Solve \( \frac{dy}{dx} = 2y \) with \( y(0) = 1 \).
Find the general solution of \( \frac{dy}{dx} = x \).
Solve \( \frac{dy}{dx} = -y \) with \( y(0) = 2 \).
Determine the solution to \( \frac{dy}{dx} = 3x^2 \) with \( y(0) = 0 \).
Solve \( \frac{dy}{dx} = y^2 \) with \( y(1) = 1 \).
Find the general solution of \( \frac{dy}{dx} = \sin x \).
Solve \( \frac{dy}{dx} = -2x \) with \( y(0) = 3 \).
Moderate Questions 🌱
Solve \( \frac{dy}{dx} = xy \) with \( y(0) = 1 \).
Find the general solution of \( \frac{dy}{dx} = e^x + y \).
Solve \( \frac{dy}{dx} = \frac{x}{y} \) with \( y(1) = 2 \).
Determine the solution to \( \frac{dy}{dx} = x^2 - y^2 \) with \( y(0) = 0 \).
Solve \( \frac{dy}{dx} = \cos x - y \) with \( y(0) = 1 \).
Find the general solution of \( \frac{dy}{dx} = \frac{1}{x} y^2 \).
Determine \( y(x) \) for \( \frac{dy}{dx} = x e^y \) with \( y(0) = 0 \).
Hard Questions 🌟
Solve \( \frac{dy}{dx} = y \ln y \) with \( y(1) = e \).
Find the general solution of \( \frac{dy}{dx} = x^2 y + x^2 \).
Determine \( y(x) \) for \( \frac{dy}{dx} = \frac{y^2 + 1}{x} \) with \( y(1) = 0 \).
Solve \( \frac{dy}{dx} = e^{x+y} \) with \( y(0) = 0 \).
Topic 2: Applications of Differential Equations (Growth, Decay, Cooling, etc.)
Easy Questions 🌱
Find the population growth model \( \frac{dP}{dt} = 0.02P \) with \( P(0) = 1000 \).
Solve the decay equation \( \frac{dN}{dt} = -0.1N \) with \( N(0) = 500 \).
Determine the cooling model \( \frac{dT}{dt} = -k(T - 20) \) with \( T(0) = 80 \) and \( k = 0.1 \).
Find \( N(t) \) for \( \frac{dN}{dt} = -0.05N \) with \( N(0) = 200 \).
Solve \( \frac{dP}{dt} = 0.03P \) with \( P(0) = 500 \).
Determine the temperature \( T(t) \) for \( \frac{dT}{dt} = -0.2(T - 25) \) with \( T(0) = 100 \).
Moderate Questions 🌱
Find the population with \( \frac{dP}{dt} = 0.01P(1000 - P) \) with \( P(0) = 100 \).
Solve the decay model \( \frac{dN}{dt} = -kN^2 \) with \( N(0) = 1 \) and \( k = 0.1 \).
Determine \( T(t) \) for \( \frac{dT}{dt} = -0.05(T - 15) \cos t \) with \( T(0) = 50 \).
Find the solution to \( \frac{dP}{dt} = 0.02P - 0.001P^2 \) with \( P(0) = 10 \).
Solve \( \frac{dN}{dt} = -0.1N + 2 \) with \( N(0) = 0 \).
Hard Questions 🌟
Determine \( P(t) \) for \( \frac{dP}{dt} = kP(1 - P/K) - h \) with \( P(0) = P_0 \), \( k = 0.1 \), \( K = 1000 \), \( h = 5 \), \( P_0 = 100 \).
Solve the mixing problem \( \frac{dC}{dt} = 0.1(20 - C) \) with \( C(0) = 0 \).
Find \( T(t) \) for \( \frac{dT}{dt} = -k(T - T_m) + A \sin t \) with \( T(0) = 30 \), \( k = 0.1 \), \( T_m = 20 \), \( A = 5 \).
Topic 3: Multivariable Functions and Partial Derivatives
Easy Questions 🌱
Find \( \frac{\partial z}{\partial x} \) for \( z = x^2 + y^2 \).
Compute \( \frac{\partial f}{\partial y} \) for \( f(x, y) = x^3 y \).
Determine \( \frac{\partial z}{\partial x} \) for \( z = \sin(x) \cos(y) \).
Find \( \frac{\partial f}{\partial y} \) for \( f(x, y) = e^{x + y} \).
Moderate Questions 🌱
Compute \( \frac{\partial^2 z}{\partial x^2} \) for \( z = x^2 y + y^3 \).
Find \( \frac{\partial f}{\partial x} \) for \( f(x, y) = \ln(x^2 + y^2) \).
Determine \( \frac{\partial^2 z}{\partial y \partial x} \) for \( z = e^{xy} \).
Compute \( \frac{\partial f}{\partial y} \) for \( f(x, y) = x^2 e^y + \sin y \).
Hard Questions 🌟
Find \( \frac{\partial^2 f}{\partial x \partial y} \) for \( f(x, y) = x^3 y^2 + \cos(xy) \).
Determine all second partial derivatives of \( f(x, y) = \frac{1}{x + y} \).
Topic 4: Gradient, Directional Derivatives, and Optimization
Easy Questions 🌱
Find the gradient of \( f(x, y) = x^2 + y^2 \) at \( (1, 1) \).
Compute the directional derivative of \( f(x, y) = x y \) at \( (2, 3) \) in the direction of \( \vec{u} = \langle 1, 1 \rangle \).
Moderate Questions 🌱
Find the gradient of \( f(x, y) = e^{x} \sin y \) at \( (0, \pi/2) \).
Compute the directional derivative of \( f(x, y) = x^2 - y^2 \) at \( (1, 1) \) in the direction \( \vec{u} = \langle 3, 4 \rangle \).
Determine the maximum rate of change of \( f(x, y) = \ln(x^2 + y^2) \) at \( (1, 1) \).
Hard Questions 🌟
Find the critical points of \( f(x, y) = x^3 + y^3 - 3xy \) and classify them.
Optimize \( f(x, y) = x^2 + y^2 \) subject to \( x + y = 4 \).
Topic 5: Double and Triple Integrals
Easy Questions 🌱
Evaluate \( \iint_R x \, dA \) where \( R = [0, 1] \times [0, 1] \).
Compute \( \iiint_E 1 \, dV \) where \( E = [0, 1] \times [0, 1] \times [0, 1] \).
Moderate Questions 🌱
Evaluate \( \iint_R (x + y) \, dA \) where \( R = [0, 2] \times [0, 1] \).
Compute \( \iiint_E x^2 \, dV \) where \( E = [0, 1] \times [0, 2] \times [0, 3] \).
Find \( \iint_R x y \, dA \) where \( R \) is the triangle with vertices \( (0, 0) \), \( (1, 0) \), \( (0, 1) \).
Hard Questions 🌟
Evaluate \( \iint_R e^{x+y} \, dA \) where \( R = [0, 1] \times [0, \ln 2] \).
Compute \( \iiint_E x y z \, dV \) where \( E \) is the region \( 0 \leq x \leq 1 \), \( 0 \leq y \leq x \), \( 0 \leq z \leq 1 \).
Topic 6: Change of Variables in Multiple Integrals (Polar, Cylindrical, Spherical)
Easy Questions 🌱
Evaluate \( \iint_R x^2 + y^2 \, dA \) in polar coordinates where \( R \) is the disk \( x^2 + y^2 \leq 1 \).
Moderate Questions 🌱
Compute \( \iint_R e^{x^2 + y^2} \, dA \) in polar coordinates where \( R \) is \( x^2 + y^2 \leq 4 \).
Evaluate \( \iiint_E z \, dV \) in cylindrical coordinates where \( E \) is \( x^2 + y^2 \leq 1 \), \( 0 \leq z \leq 1 \).
Hard Questions 🌟
Find \( \iiint_E x^2 + y^2 + z^2 \, dV \) in spherical coordinates where \( E \) is the sphere \( x^2 + y^2 + z^2 \leq 1 \).
Topic 7: Sequences and Their Limits
Easy Questions 🌱
Find the limit of \( a_n = \frac{1}{n} \) as \( n \to \infty \).
Determine if \( a_n = 2 + \frac{1}{n} \) converges and find its limit.
Moderate Questions 🌱
Find the limit of \( a_n = \frac{n^2}{n + 1} \) as \( n \to \infty \).
Determine the limit of \( a_n = \frac{\sin n}{n} \) as \( n \to \infty \).
Hard Questions 🌟
Prove that \( a_n = \frac{n!}{n^n} \to 0 \) as \( n \to \infty \).
Topic 8: Infinite Series and Convergence Tests
Easy Questions 🌱
Determine if \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges.
Test \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) for convergence.
Moderate Questions 🌱
Check convergence of \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \) using the partial fraction method.
Determine if \( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \) converges.
Hard Questions 🌟
Test \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n \ln n} \) for convergence.
Topic 9: Power Series and Taylor Series
Easy Questions 🌱
Find the radius of convergence of \( \sum_{n=0}^{\infty} x^n \).
Compute the Taylor series of \( f(x) = e^x \) at \( x = 0 \) up to \( x^3 \).
Moderate Questions 🌱
Determine the radius of convergence of \( \sum_{n=1}^{\infty} \frac{x^n}{n} \).
Find the Taylor series of \( f(x) = \sin x \) at \( x = 0 \) up to \( x^4 \).
Hard Questions 🌟
Compute the Taylor series of \( f(x) = \ln(1 + x) \) at \( x = 0 \) and find its radius of convergence.