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

Attempt · Explain · Animate

Undergraduate practice chapters

1. Undergraduate · Chapter 161

   Euler stability · Decay rate a=1.75; 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=1.75; Step size h=0.8.

   Question · Worked solution · Animated example

   Open practice chapter

2. Undergraduate · Chapter 162

   Newton root iteration · Number whose root is sought A=6; Positive starting guess b=3.5

   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=6; Positive starting guess b=3.5.

   Question · Worked solution · Animated example

   Open practice chapter

3. Undergraduate · Chapter 163

   Two-state Markov chains · Transition 1 → 2: p=0.2; 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=0.2; Transition 2 → 1: q=0.4.

   Question · Worked solution · Animated example

   Open practice chapter

4. Undergraduate · Chapter 164

   Orthogonal projections · Vector horizontal a=1; 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=1; Vector vertical b=1.5.

   Question · Worked solution · Animated example

   Open practice chapter

5. Undergraduate · Chapter 165

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

   Question · Worked solution · Animated example

   Open practice chapter

6. Undergraduate · Chapter 166

   Logistic differential equations · Capacity K=5.5; Growth rate r=0.2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5.5; Growth rate r=0.2.

   Question · Worked solution · Animated example

   Open practice chapter

7. Undergraduate · Chapter 167

   Euler stability · Decay rate a=2; Step size h=0.9

   Use n=6 updates, corresponding to time t=nh=5.4. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=2; Step size h=0.9.

   Question · Worked solution · Animated example

   Open practice chapter

8. Undergraduate · Chapter 168

   Newton root iteration · Number whose root is sought A=7; Positive starting guess b=4

   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=7; Positive starting guess b=4.

   Question · Worked solution · Animated example

   Open practice chapter

9. Undergraduate · Chapter 169

   Two-state Markov chains · Transition 1 → 2: p=0.3; 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=0.3; Transition 2 → 1: q=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

10. Undergraduate · Chapter 170

    Orthogonal projections · Vector horizontal a=1.5; Vector vertical b=2

    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.

    Question · Worked solution · Animated example

    Open practice chapter

11. Undergraduate · Chapter 171

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

    Question · Worked solution · Animated example

    Open practice chapter

12. Undergraduate · Chapter 172

    Matrix conditioning · Small diagonal entry a=0.1; 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=0.1; Final data error b=0.3.

    Question · Worked solution · Animated example

    Open practice chapter

13. Undergraduate · Chapter 173

    Directional derivatives · x² coefficient a=2; y² coefficient b=0.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=2; y² coefficient b=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

14. Undergraduate · Chapter 174

    Logistic differential equations · Capacity K=6; Growth rate r=0.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=6; Growth rate r=0.4.

    Question · Worked solution · Animated example

    Open practice chapter

15. Undergraduate · Chapter 175

    Harmonic oscillators · Angular frequency ω=2; Initial displacement b=1

    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; Initial displacement b=1.

    Question · Worked solution · Animated example

    Open practice chapter

16. Undergraduate · Chapter 176

    Euler stability · Decay rate a=0.5; Step size h=0.4

    Use n=6 updates, corresponding to time t=nh=2.4. 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=0.4.

    Question · Worked solution · Animated example

    Open practice chapter

17. Undergraduate · Chapter 177

    Newton root iteration · Number whose root is sought A=8; Positive starting guess b=4.5

    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=8; Positive starting guess b=4.5.

    Question · Worked solution · Animated example

    Open practice chapter

18. Undergraduate · Chapter 178

    Orthogonal projections · Vector horizontal a=2; 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=2; Vector vertical b=2.5.

    Question · Worked solution · Animated example

    Open practice chapter

19. Undergraduate · Chapter 179

    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

20. Undergraduate · Chapter 180

    Matrix conditioning · Small diagonal entry a=0.15; Final data error b=0.4

    Use the current perturbation δ=0.24, so Δb=(0,0.24). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.15; Final data error b=0.4.

    Question · Worked solution · Animated example

    Open practice chapter