Topic 1: Introduction to Differential Equations (Separable and First-Order ODEs)
1\( y = e^{2x} \)
2\( y = \frac{x^2}{2} + C \)
3\( y = 2e^{-x} \)
4\( y = x^3 \)
5\( y = \frac{1}{1 - x} \)
6\( y = -\cos x + C \)
7\( y = 3 - x^2 \)
8\( y = e^{x^2/2} \)
9\( y = \frac{1}{1 - x} \)
10\( y = e^{x - 1} + \frac{e^x}{e - 1} \)
11\( y = \frac{1}{1 - e^{x^2/2}} \)
12\( y = \frac{e^x}{e^x - 1} \)
13\( y = -\frac{x^2}{2} \)
14\( y = e^{x^2} \)
Topic 2: Applications of Differential Equations (Growth, Decay, Cooling, etc.)
1\( P = 1000 e^{0.02t} \)
2\( N = 500 e^{-0.1t} \)
3\( T = 20 + 60 e^{-0.1t} \)
4\( N = 200 e^{-0.05t} \)
5\( P = 500 e^{0.03t} \)
6\( T = 25 + 75 e^{-0.2t} \)
7\( P = \frac{1000}{1 + 9e^{-0.01t}} \)
8\( N = \frac{1}{0.1t + 1} \)
9\( T = 15 + 35 e^{-0.05t} \cos t \)
10\( P = \frac{20}{1 + 1.9e^{-0.02t}} \)
11\( N = \frac{20}{20 - C + 20e^{-0.1t}} \)
12\( T = 20 + 10 e^{-0.1t} + 50 \sin t \)
Topic 3: Multivariable Functions and Partial Derivatives
1\( 2x \)
2\( x^3 \)
3\( \cos(x) \cos(y) \)
4\( e^{x + y} \)
5\( 2xy \)
6\( \frac{2x}{x^2 + y^2} \)
7\( y e^{xy} \)
8\( x^2 e^y + \cos y \)
9\( 6x^2 y^2 - y \sin(xy) \)
10\( \frac{-1}{(x + y)^2} \) (all second derivatives)
Topic 4: Gradient, Directional Derivatives, and Optimization
1\( \langle 2, 2 \rangle \)
2\( \frac{5}{\sqrt{2}} \)
3\( \langle 1, 0 \rangle \)
4\( -\frac{5}{5} \)
5\( \sqrt{2} \)
6\( (0, 0), (1, 1), (1, -1) \) (saddle, min, max)
7\( (2, 2), 8 \)
Topic 5: Double and Triple Integrals
1\( \frac{1}{2} \)
2\( 1 \)
3\( 1 \)
4\( 6 \)
5\( \frac{1}{2} \)
6\( \frac{1}{2} (e^2 - 1) \)
7\( \frac{1}{4} \)
Topic 6: Change of Variables in Multiple Integrals (Polar, Cylindrical, Spherical)
1\( \frac{\pi}{2} \)
2\( \pi (e^4 - 1) \)
3\( \frac{\pi}{2} \)
4\( \frac{\pi}{3} \)
Topic 7: Sequences and Their Limits
1\( 0 \)
2\( 2 \)
3\( \infty \)
4\( 0 \)
5\( 0 \) (by squeeze theorem)
Topic 8: Infinite Series and Convergence Tests
1Converges (p-series, \( p = 2 > 1 \))
2Converges (geometric, \( |r| = \frac{1}{2} < 1 \))
3Converges (telescoping, sum = 1)
4Diverges (integral test)
5Converges (alternating series test)
Topic 9: Power Series and Taylor Series
1\( 1 \)
2\( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \)
3\( 1 \)
4\( x - \frac{x^3}{6} + \frac{x^5}{120} \)
5\( x - \frac{x^2}{2} + \frac{x^3}{3} \), radius = 1