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

Attempt · Explain · Animate

Undergraduate practice chapters

1. Undergraduate · Chapter 581

   Newton root iteration · Number whose root is sought A=15; Positive starting guess b=1.5

   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=15; Positive starting guess b=1.5.

   Question · Worked solution · Animated example

   Open practice chapter

2. Undergraduate · Chapter 582

   Two-state Markov chains · Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.4

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.4.

   Question · Worked solution · Animated example

   Open practice chapter

3. Undergraduate · Chapter 583

   Orthogonal projections · Vector horizontal a=-3; Vector vertical b=0

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-3; Vector vertical b=0.

   Question · Worked solution · Animated example

   Open practice chapter

4. Undergraduate · Chapter 584

   Diagonal eigenmodes · First eigenvalue a=-1.5; Second eigenvalue b=2

   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=2.

   Question · Worked solution · Animated example

   Open practice chapter

5. Undergraduate · Chapter 585

   Matrix conditioning · Small diagonal entry a=0.9; Final data error b=0.1

   Use the current perturbation δ=0.06, so Δb=(0,0.06). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.9; Final data error b=0.1.

   Question · Worked solution · Animated example

   Open practice chapter

6. Undergraduate · Chapter 586

   Logistic differential equations · Capacity K=1; Growth rate r=1.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1; Growth rate r=1.6.

   Question · Worked solution · Animated example

   Open practice chapter

7. Undergraduate · Chapter 587

   Euler stability · Decay rate a=0.5; 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=0.5; Step size h=1.2.

   Question · Worked solution · Animated example

   Open practice chapter

8. Undergraduate · Chapter 588

   Two-state Markov chains · Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.5

   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=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

9. Undergraduate · Chapter 589

   Orthogonal projections · Vector horizontal a=-2.5; 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=-2.5; Vector vertical b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

10. Undergraduate · Chapter 590

    Diagonal eigenmodes · First eigenvalue a=-1; Second eigenvalue b=2.5

    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=2.5.

    Question · Worked solution · Animated example

    Open practice chapter

11. Undergraduate · Chapter 591

    Matrix conditioning · Small diagonal entry a=0.95; Final data error b=0.2

    Use the current perturbation δ=0.12, so Δb=(0,0.12). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.95; Final data error b=0.2.

    Question · Worked solution · Animated example

    Open practice chapter

12. Undergraduate · Chapter 592

    Logistic differential equations · Capacity K=1.5; Growth rate r=1.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1.5; Growth rate r=1.8.

    Question · Worked solution · Animated example

    Open practice chapter

13. Undergraduate · Chapter 593

    Euler stability · Decay rate a=0.75; 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=0.75; Step size h=1.3.

    Question · Worked solution · Animated example

    Open practice chapter

14. Undergraduate · Chapter 594

    Newton root iteration · Number whose root is sought A=1; Positive starting guess b=3

    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=1; Positive starting guess b=3.

    Question · Worked solution · Animated example

    Open practice chapter

15. Undergraduate · Chapter 595

    Two-state Markov chains · Transition 1 → 2: p=0.7; 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=0.7; Transition 2 → 1: q=0.6.

    Question · Worked solution · Animated example

    Open practice chapter

16. Undergraduate · Chapter 596

    Orthogonal projections · Vector horizontal a=-2; 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=-2; Vector vertical b=1.

    Question · Worked solution · Animated example

    Open practice chapter

17. Undergraduate · Chapter 597

    Diagonal eigenmodes · First eigenvalue a=-0.5; Second eigenvalue b=3

    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=-0.5; Second eigenvalue b=3.

    Question · Worked solution · Animated example

    Open practice chapter

18. Undergraduate · Chapter 598

    Matrix conditioning · Small diagonal entry a=1; Final data error b=0.3

    Use the current perturbation δ=0.18, so Δb=(0,0.18). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=1; Final data error b=0.3.

    Question · Worked solution · Animated example

    Open practice chapter

19. Undergraduate · Chapter 599

    Logistic differential equations · Capacity K=2; Growth rate r=2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2; Growth rate r=2.

    Question · Worked solution · Animated example

    Open practice chapter

20. Undergraduate · Chapter 600

    Euler stability · Decay rate a=1; Step size h=1.4

    Use n=6 updates, corresponding to time t=nh=8.4. 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=1.4.

    Question · Worked solution · Animated example

    Open practice chapter