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

Attempt · Explain · Animate

Undergraduate practice chapters

1. Undergraduate · Chapter 221

   Newton root iteration · Number whose root is sought A=13; 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=13; Positive starting guess b=7.

   Question · Worked solution · Animated example

   Open practice chapter

2. Undergraduate · Chapter 222

   Two-state Markov chains · Transition 1 → 2: p=0.9; 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=0.9; Transition 2 → 1: q=0.2.

   Question · Worked solution · Animated example

   Open practice chapter

3. Undergraduate · Chapter 223

   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

4. Undergraduate · Chapter 224

   Diagonal eigenmodes · First eigenvalue a=1; 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; Second eigenvalue b=2.

   Question · Worked solution · Animated example

   Open practice chapter

5. Undergraduate · Chapter 225

   Matrix conditioning · Small diagonal entry a=0.4; Final data error b=0.9

   Use the current perturbation δ=0.54, so Δb=(0,0.54). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.4; Final data error b=0.9.

   Question · Worked solution · Animated example

   Open practice chapter

6. Undergraduate · Chapter 226

   Logistic differential equations · Capacity K=3.5; Growth rate r=1.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3.5; Growth rate r=1.6.

   Question · Worked solution · Animated example

   Open practice chapter

7. Undergraduate · Chapter 227

   Newton root iteration · Number whose root is sought A=14; 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=14; Positive starting guess b=7.5.

   Question · Worked solution · Animated example

   Open practice chapter

8. Undergraduate · Chapter 228

   Two-state Markov chains · Transition 1 → 2: p=0.1; 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=0.1; Transition 2 → 1: q=0.4.

   Question · Worked solution · Animated example

   Open practice chapter

9. Undergraduate · Chapter 229

   Orthogonal projections · Vector horizontal a=-1.5; 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=-1.5; Vector vertical b=-0.5.

   Question · Worked solution · Animated example

   Open practice chapter

10. Undergraduate · Chapter 230

    Diagonal eigenmodes · First eigenvalue a=1.5; 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=1.5; Second eigenvalue b=2.5.

    Question · Worked solution · Animated example

    Open practice chapter

11. Undergraduate · Chapter 231

    Matrix conditioning · Small diagonal entry a=0.45; Final data error b=1

    Use the current perturbation δ=0.6, so Δb=(0,0.6). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.45; Final data error b=1.

    Question · Worked solution · Animated example

    Open practice chapter

12. Undergraduate · Chapter 232

    Directional derivatives · x² coefficient a=2.5; y² coefficient b=1.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=2.5; y² coefficient b=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

13. Undergraduate · Chapter 233

    Harmonic oscillators · Angular frequency ω=2.5; Initial displacement b=2

    Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=2.5; Initial displacement b=2.

    Question · Worked solution · Animated example

    Open practice chapter

14. Undergraduate · Chapter 234

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

    Use n=6 updates, corresponding to time t=nh=3. 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=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

15. Undergraduate · Chapter 235

    Newton root iteration · Number whose root is sought A=15; 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=15; Positive starting guess b=8.

    Question · Worked solution · Animated example

    Open practice chapter

16. Undergraduate · Chapter 236

    Two-state Markov chains · Transition 1 → 2: p=0.2; 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=0.2; Transition 2 → 1: q=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

17. Undergraduate · Chapter 237

    Orthogonal projections · Vector horizontal a=-1; 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=-1; Vector vertical b=0.

    Question · Worked solution · Animated example

    Open practice chapter

18. Undergraduate · Chapter 238

    Diagonal eigenmodes · First eigenvalue a=2; 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=2; Second eigenvalue b=3.

    Question · Worked solution · Animated example

    Open practice chapter

19. Undergraduate · Chapter 239

    Matrix conditioning · Small diagonal entry a=0.5; 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=0.5; Final data error b=0.1.

    Question · Worked solution · Animated example

    Open practice chapter

20. Undergraduate · Chapter 240

    Directional derivatives · x² coefficient a=3; y² coefficient b=2

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=3; y² coefficient b=2.

    Question · Worked solution · Animated example

    Open practice chapter