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

Attempt · Explain · Animate

Graduate practice chapters

1. Graduate · Chapter 621

   Contraction estimates · Contraction slope q=0.1; 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=0.1; Constant b=0.5.

   Question · Worked solution · Animated example

   Open practice chapter

2. Graduate · Chapter 622

   Smoothing a weak derivative · Smoothing width ε=0.7; 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=0.7; Corner location b=1.

   Question · Worked solution · Animated example

   Open practice chapter

3. Graduate · Chapter 623

   Coercivity in a finite model · First diagonal entry a=0.8; Second diagonal entry b=2

   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=2.

   Question · Worked solution · Animated example

   Open practice chapter

4. Graduate · Chapter 624

   Critical norm scaling · Final spatial factor λ=1.25; Lebesgue exponent p=6

   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=6.

   Question · Worked solution · Animated example

   Open practice chapter

5. Graduate · Chapter 625

   Conditional expectation by groups · First group baseline a=0.5; 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=0.5; Second group baseline b=-1.

   Question · Worked solution · Animated example

   Open practice chapter

6. Graduate · Chapter 626

   Smoothing a weak derivative · Smoothing width ε=0.75; 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=0.75; Corner location b=-1.

   Question · Worked solution · Animated example

   Open practice chapter

7. Graduate · Chapter 627

   Coercivity in a finite model · First diagonal entry a=1; Second diagonal entry b=2.2

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1; Second diagonal entry b=2.2.

   Question · Worked solution · Animated example

   Open practice chapter

8. Graduate · Chapter 628

   Critical norm scaling · Final spatial factor λ=1.5; Lebesgue exponent p=1

   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=1.

   Question · Worked solution · Animated example

   Open practice chapter

9. Graduate · Chapter 629

   Conditional expectation by groups · First group baseline a=1; 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; Second group baseline b=-0.5.

   Question · Worked solution · Animated example

   Open practice chapter

10. Graduate · Chapter 630

    Contraction estimates · Contraction slope q=0.3; 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=0.3; Constant b=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

11. Graduate · Chapter 631

    Smoothing a weak derivative · Smoothing width ε=0.8; 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=0.8; Corner location b=-0.75.

    Question · Worked solution · Animated example

    Open practice chapter

12. Graduate · Chapter 632

    Coercivity in a finite model · First diagonal entry a=1.2; Second diagonal entry b=2.4

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.2; Second diagonal entry b=2.4.

    Question · Worked solution · Animated example

    Open practice chapter

13. Graduate · Chapter 633

    Critical norm scaling · Final spatial factor λ=1.75; Lebesgue exponent p=1.5

    Evaluate at the current scale λ=1.45. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.75; Lebesgue exponent p=1.5.

    Question · Worked solution · Animated example

    Open practice chapter

14. Graduate · Chapter 634

    Conditional expectation by groups · First group baseline a=1.5; 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=1.5; Second group baseline b=0.

    Question · Worked solution · Animated example

    Open practice chapter

15. Graduate · Chapter 635

    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

16. Graduate · Chapter 636

    Smoothing a weak derivative · Smoothing width ε=0.85; 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=0.85; Corner location b=-0.5.

    Question · Worked solution · Animated example

    Open practice chapter

17. Graduate · Chapter 637

    Coercivity in a finite model · First diagonal entry a=1.4; Second diagonal entry b=2.6

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.4; Second diagonal entry b=2.6.

    Question · Worked solution · Animated example

    Open practice chapter

18. Graduate · Chapter 638

    Critical norm scaling · Final spatial factor λ=2; Lebesgue exponent p=2

    Evaluate at the current scale λ=1.6. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2; Lebesgue exponent p=2.

    Question · Worked solution · Animated example

    Open practice chapter

19. Graduate · Chapter 639

    Conditional expectation by groups · First group baseline a=2; 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; Second group baseline b=0.5.

    Question · Worked solution · Animated example

    Open practice chapter

20. Graduate · Chapter 640

    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