10,000 ways to practice.
One idea at a time.

Attempt · Explain · Animate

Undergraduate practice chapters

1. Undergraduate · Chapter 341

   Two-state Markov chains · Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.1

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.1.

   Question · Worked solution · Animated example

   Open practice chapter

2. Undergraduate · Chapter 342

   Orthogonal projections · Vector horizontal a=-1; Vector vertical b=0.5

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-1; Vector vertical b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

3. Undergraduate · Chapter 343

   Diagonal eigenmodes · First eigenvalue a=3; Second eigenvalue b=-1

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=3; Second eigenvalue b=-1.

   Question · Worked solution · Animated example

   Open practice chapter

4. Undergraduate · Chapter 344

   Matrix conditioning · Small diagonal entry a=0.15; Final data error b=0.5

   Use the current perturbation δ=0.3, so Δb=(0,0.3). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.15; Final data error b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

5. Undergraduate · Chapter 345

   Logistic differential equations · Capacity K=5.5; Growth rate r=0.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5.5; Growth rate r=0.6.

   Question · Worked solution · Animated example

   Open practice chapter

6. Undergraduate · Chapter 346

   Harmonic oscillators · Angular frequency ω=0.5; Initial displacement b=1.5

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=0.5; Initial displacement b=1.5.

   Question · Worked solution · Animated example

   Open practice chapter

7. Undergraduate · Chapter 347

   Euler stability · Decay rate a=0.5; Step size h=0.7

   Use n=6 updates, corresponding to time t=nh=4.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.5; Step size h=0.7.

   Question · Worked solution · Animated example

   Open practice chapter

8. Undergraduate · Chapter 348

   Newton root iteration · Number whose root is sought A=13; Positive starting guess b=7.5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=13; Positive starting guess b=7.5.

   Question · Worked solution · Animated example

   Open practice chapter

9. Undergraduate · Chapter 349

   Two-state Markov chains · Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.2

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.2.

   Question · Worked solution · Animated example

   Open practice chapter

10. Undergraduate · Chapter 350

    Orthogonal projections · Vector horizontal a=-0.5; Vector vertical b=1

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-0.5; Vector vertical b=1.

    Question · Worked solution · Animated example

    Open practice chapter

11. Undergraduate · Chapter 351

    Diagonal eigenmodes · First eigenvalue a=-2; Second eigenvalue b=0

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-2; Second eigenvalue b=0.

    Question · Worked solution · Animated example

    Open practice chapter

12. Undergraduate · Chapter 352

    Matrix conditioning · Small diagonal entry a=0.2; Final data error b=0.6

    Use the current perturbation δ=0.36, so Δb=(0,0.36). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.2; Final data error b=0.6.

    Question · Worked solution · Animated example

    Open practice chapter

13. Undergraduate · Chapter 353

    Logistic differential equations · Capacity K=6; Growth rate r=0.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=6; Growth rate r=0.8.

    Question · Worked solution · Animated example

    Open practice chapter

14. Undergraduate · Chapter 354

    Euler stability · Decay rate a=0.75; Step size h=0.8

    Use n=6 updates, corresponding to time t=nh=4.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.75; Step size h=0.8.

    Question · Worked solution · Animated example

    Open practice chapter

15. Undergraduate · Chapter 355

    Newton root iteration · Number whose root is sought A=14; Positive starting guess b=8

    Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=14; Positive starting guess b=8.

    Question · Worked solution · Animated example

    Open practice chapter

16. Undergraduate · Chapter 356

    Two-state Markov chains · Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.3

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.3.

    Question · Worked solution · Animated example

    Open practice chapter

17. Undergraduate · Chapter 357

    Orthogonal projections · Vector horizontal a=0; Vector vertical b=1.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=0; Vector vertical b=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

18. Undergraduate · Chapter 358

    Diagonal eigenmodes · First eigenvalue a=-1.5; Second eigenvalue b=0.5

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-1.5; Second eigenvalue b=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

19. Undergraduate · Chapter 359

    Matrix conditioning · Small diagonal entry a=0.25; Final data error b=0.7

    Use the current perturbation δ=0.42, so Δb=(0,0.42). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.25; Final data error b=0.7.

    Question · Worked solution · Animated example

    Open practice chapter

20. Undergraduate · Chapter 360

    Logistic differential equations · Capacity K=1; Growth rate r=1

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1; Growth rate r=1.

    Question · Worked solution · Animated example

    Open practice chapter