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

Attempt · Explain · Animate

Graduate practice chapters

1. Graduate · Chapter 541

   Conditional expectation by groups · First group baseline a=2.5; Second group baseline b=0

   Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=2.5; Second group baseline b=0.

   Question · Worked solution · Animated example

   Open practice chapter

2. Graduate · Chapter 542

   Contraction estimates · Contraction slope q=0.1; Constant b=0

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=0.1; Constant b=0.

   Question · Worked solution · Animated example

   Open practice chapter

3. Graduate · Chapter 543

   Smoothing a weak derivative · Smoothing width ε=0.85; Corner location b=-0.75

   Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.85; Corner location b=-0.75.

   Question · Worked solution · Animated example

   Open practice chapter

4. Graduate · Chapter 544

   Coercivity in a finite model · First diagonal entry a=0.4; Second diagonal entry b=1.4

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.4; Second diagonal entry b=1.4.

   Question · Worked solution · Animated example

   Open practice chapter

5. Graduate · Chapter 545

   Rare events and convergence · Magnitude exponent a=2; Rarity exponent b=1.75

   Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=2; Rarity exponent b=1.75.

   Question · Worked solution · Animated example

   Open practice chapter

6. Graduate · Chapter 546

   Conditional expectation by groups · First group baseline a=3; Second group baseline b=0.5

   Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=3; Second group baseline b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

7. Graduate · Chapter 547

   Contraction estimates · Contraction slope q=0.2; Constant b=0.5

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=0.2; Constant b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

8. Graduate · Chapter 548

   Smoothing a weak derivative · Smoothing width ε=0.9; Corner location b=-0.5

   Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.9; Corner location b=-0.5.

   Question · Worked solution · Animated example

   Open practice chapter

9. Graduate · Chapter 549

   Coercivity in a finite model · First diagonal entry a=0.6; Second diagonal entry b=1.6

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.6; Second diagonal entry b=1.6.

   Question · Worked solution · Animated example

   Open practice chapter

10. Graduate · Chapter 550

    Critical norm scaling · Final spatial factor λ=0.75; Lebesgue exponent p=4.5

    Evaluate at the current scale λ=0.85. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.75; Lebesgue exponent p=4.5.

    Question · Worked solution · Animated example

    Open practice chapter

11. Graduate · Chapter 551

    Conditional expectation by groups · First group baseline a=-2; Second group baseline b=1.5

    Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-2; Second group baseline b=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

12. Graduate · Chapter 552

    Contraction estimates · Contraction slope q=0.3; Constant b=1

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=0.3; Constant b=1.

    Question · Worked solution · Animated example

    Open practice chapter

13. Graduate · Chapter 553

    Smoothing a weak derivative · Smoothing width ε=0.95; Corner location b=-0.25

    Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.95; Corner location b=-0.25.

    Question · Worked solution · Animated example

    Open practice chapter

14. Graduate · Chapter 554

    Coercivity in a finite model · First diagonal entry a=0.8; Second diagonal entry b=1.8

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.8; Second diagonal entry b=1.8.

    Question · Worked solution · Animated example

    Open practice chapter

15. Graduate · Chapter 555

    Critical norm scaling · Final spatial factor λ=1.25; Lebesgue exponent p=5.5

    Evaluate at the current scale λ=1.15. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.25; Lebesgue exponent p=5.5.

    Question · Worked solution · Animated example

    Open practice chapter

16. Graduate · Chapter 556

    Conditional expectation by groups · First group baseline a=-1.5; Second group baseline b=2

    Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-1.5; Second group baseline b=2.

    Question · Worked solution · Animated example

    Open practice chapter

17. Graduate · Chapter 557

    Contraction estimates · Contraction slope q=0.4; Constant b=1.5

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=0.4; Constant b=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

18. Graduate · Chapter 558

    Critical norm scaling · Final spatial factor λ=1.5; Lebesgue exponent p=6

    Evaluate at the current scale λ=1.3. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.5; Lebesgue exponent p=6.

    Question · Worked solution · Animated example

    Open practice chapter

19. Graduate · Chapter 559

    Conditional expectation by groups · First group baseline a=-1; Second group baseline b=2.5

    Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-1; Second group baseline b=2.5.

    Question · Worked solution · Animated example

    Open practice chapter

20. Graduate · Chapter 560

    Contraction estimates · Contraction slope q=0.5; Constant b=2

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=0.5; Constant b=2.

    Question · Worked solution · Animated example

    Open practice chapter