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

Attempt · Explain · Animate

Graduate practice chapters

1. Graduate · Chapter 581

   Coercivity in a finite model · First diagonal entry a=2; Second diagonal entry b=3

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2; Second diagonal entry b=3.

   Question · Worked solution · Animated example

   Open practice chapter

2. Graduate · Chapter 582

   Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=3

   Evaluate at the current scale λ=2.05. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.75; Lebesgue exponent p=3.

   Question · Worked solution · Animated example

   Open practice chapter

3. Graduate · Chapter 583

   Conditional expectation by groups · First group baseline a=1.5; 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=1.5; Second group baseline b=-0.5.

   Question · Worked solution · Animated example

   Open practice chapter

4. Graduate · Chapter 584

   Contraction estimates · Contraction slope q=-0.7; 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=-0.7; Constant b=1.

   Question · Worked solution · Animated example

   Open practice chapter

5. Graduate · Chapter 585

   Smoothing a weak derivative · Smoothing width ε=0.3; Corner location b=-1

   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=0.3; Corner location b=-1.

   Question · Worked solution · Animated example

   Open practice chapter

6. Graduate · Chapter 586

   Coercivity in a finite model · First diagonal entry a=2.2; Second diagonal entry b=0.2

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.2; Second diagonal entry b=0.2.

   Question · Worked solution · Animated example

   Open practice chapter

7. Graduate · Chapter 587

   Contraction estimates · Contraction slope q=-0.6; 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=-0.6; Constant b=1.5.

   Question · Worked solution · Animated example

   Open practice chapter

8. Graduate · Chapter 588

   Smoothing a weak derivative · Smoothing width ε=0.35; 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=0.35; Corner location b=-0.75.

   Question · Worked solution · Animated example

   Open practice chapter

9. Graduate · Chapter 589

   Coercivity in a finite model · First diagonal entry a=2.4; Second diagonal entry b=0.4

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.4; Second diagonal entry b=0.4.

   Question · Worked solution · Animated example

   Open practice chapter

10. Graduate · Chapter 590

    Critical norm scaling · Final spatial factor λ=3.25; Lebesgue exponent p=4

    Evaluate at the current scale λ=2.35. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.25; Lebesgue exponent p=4.

    Question · Worked solution · Animated example

    Open practice chapter

11. Graduate · Chapter 591

    Conditional expectation by groups · First group baseline a=2.5; 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=2.5; Second group baseline b=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

12. Graduate · Chapter 592

    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

13. Graduate · Chapter 593

    Smoothing a weak derivative · Smoothing width ε=0.4; 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=0.4; Corner location b=-0.5.

    Question · Worked solution · Animated example

    Open practice chapter

14. Graduate · Chapter 594

    Coercivity in a finite model · First diagonal entry a=2.6; Second diagonal entry b=0.6

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.6; Second diagonal entry b=0.6.

    Question · Worked solution · Animated example

    Open practice chapter

15. Graduate · Chapter 595

    Critical norm scaling · Final spatial factor λ=3.5; Lebesgue exponent p=4.5

    Evaluate at the current scale λ=2.5. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.5; Lebesgue exponent p=4.5.

    Question · Worked solution · Animated example

    Open practice chapter

16. Graduate · Chapter 596

    Conditional expectation by groups · First group baseline a=3; Second group baseline b=1

    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=1.

    Question · Worked solution · Animated example

    Open practice chapter

17. Graduate · Chapter 597

    Contraction estimates · Contraction slope q=-0.4; 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=-0.4; Constant b=-2.

    Question · Worked solution · Animated example

    Open practice chapter

18. Graduate · Chapter 598

    Smoothing a weak derivative · Smoothing width ε=0.45; 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=0.45; Corner location b=-0.25.

    Question · Worked solution · Animated example

    Open practice chapter

19. Graduate · Chapter 599

    Coercivity in a finite model · First diagonal entry a=2.8; Second diagonal entry b=0.8

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.8; Second diagonal entry b=0.8.

    Question · Worked solution · Animated example

    Open practice chapter

20. Graduate · Chapter 600

    Critical norm scaling · Final spatial factor λ=3.75; Lebesgue exponent p=5

    Evaluate at the current scale λ=2.65. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.75; Lebesgue exponent p=5.

    Question · Worked solution · Animated example

    Open practice chapter