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

Attempt · Explain · Animate

Undergraduate practice chapters

1. Undergraduate · Chapter 541

   Euler stability · Decay rate a=0.5; Step size h=1.1

   Use n=6 updates, corresponding to time t=nh=6.6. 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=1.1.

   Question · Worked solution · Animated example

   Open practice chapter

2. Undergraduate · Chapter 542

   Newton root iteration · Number whose root is sought A=9; Positive starting guess b=6.5

   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=9; Positive starting guess b=6.5.

   Question · Worked solution · Animated example

   Open practice chapter

3. Undergraduate · Chapter 543

   Two-state Markov chains · Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.6

   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=0.6.

   Question · Worked solution · Animated example

   Open practice chapter

4. Undergraduate · Chapter 544

   Diagonal eigenmodes · First eigenvalue a=1; Second eigenvalue b=-1.5

   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; Second eigenvalue b=-1.5.

   Question · Worked solution · Animated example

   Open practice chapter

5. Undergraduate · Chapter 545

   Matrix conditioning · Small diagonal entry a=0.6; 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=0.6; Final data error b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

6. Undergraduate · Chapter 546

   Logistic differential equations · Capacity K=3.5; Growth rate r=0.4

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3.5; Growth rate r=0.4.

   Question · Worked solution · Animated example

   Open practice chapter

7. Undergraduate · Chapter 547

   Euler stability · Decay rate a=0.75; Step size h=1.2

   Use n=6 updates, corresponding to time t=nh=7.2. 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=1.2.

   Question · Worked solution · Animated example

   Open practice chapter

8. Undergraduate · Chapter 548

   Newton root iteration · Number whose root is sought A=10; Positive starting guess b=7

   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=10; Positive starting guess b=7.

   Question · Worked solution · Animated example

   Open practice chapter

9. Undergraduate · Chapter 549

   Two-state Markov chains · Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.7

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.7.

   Question · Worked solution · Animated example

   Open practice chapter

10. Undergraduate · Chapter 550

    Orthogonal projections · Vector horizontal a=1; Vector vertical b=-3

    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=-3.

    Question · Worked solution · Animated example

    Open practice chapter

11. Undergraduate · Chapter 551

    Diagonal eigenmodes · First eigenvalue a=1.5; 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=1.5; Second eigenvalue b=-1.

    Question · Worked solution · Animated example

    Open practice chapter

12. Undergraduate · Chapter 552

    Matrix conditioning · Small diagonal entry a=0.65; 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=0.65; Final data error b=0.6.

    Question · Worked solution · Animated example

    Open practice chapter

13. Undergraduate · Chapter 553

    Logistic differential equations · Capacity K=4; Growth rate r=0.6

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4; Growth rate r=0.6.

    Question · Worked solution · Animated example

    Open practice chapter

14. Undergraduate · Chapter 554

    Euler stability · Decay rate a=1; Step size h=1.3

    Use n=6 updates, corresponding to time t=nh=7.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1; Step size h=1.3.

    Question · Worked solution · Animated example

    Open practice chapter

15. Undergraduate · Chapter 555

    Newton root iteration · Number whose root is sought A=11; 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=11; Positive starting guess b=7.5.

    Question · Worked solution · Animated example

    Open practice chapter

16. Undergraduate · Chapter 556

    Two-state Markov chains · Transition 1 → 2: p=0.1; Transition 2 → 1: q=0.9

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.1; Transition 2 → 1: q=0.9.

    Question · Worked solution · Animated example

    Open practice chapter

17. Undergraduate · Chapter 557

    Orthogonal projections · Vector horizontal a=1.5; Vector vertical b=-2.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=1.5; Vector vertical b=-2.5.

    Question · Worked solution · Animated example

    Open practice chapter

18. Undergraduate · Chapter 558

    Diagonal eigenmodes · First eigenvalue a=2; 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=2; Second eigenvalue b=-0.5.

    Question · Worked solution · Animated example

    Open practice chapter

19. Undergraduate · Chapter 559

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

    Question · Worked solution · Animated example

    Open practice chapter

20. Undergraduate · Chapter 560

    Logistic differential equations · Capacity K=4.5; Growth rate r=0.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4.5; Growth rate r=0.8.

    Question · Worked solution · Animated example

    Open practice chapter