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Undergraduate practice chapters
641–650 of 650 matching chapters · Page 33 of 33
Clear filtersUndergraduate · Chapter 641
Newton root iteration · Number whose root is sought A=9; 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=9; Positive starting guess b=7.
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Open practice chapterUndergraduate · Chapter 642
Orthogonal projections · Vector horizontal a=2; 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=2; Vector vertical b=-1.5.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 643
Diagonal eigenmodes · First eigenvalue a=-2; 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=-2; Second eigenvalue b=2.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 644
Matrix conditioning · Small diagonal entry a=0.4; Final data error b=0.2
Use the current perturbation δ=0.12, so Δb=(0,0.12). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.4; Final data error b=0.2.
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Open practice chapterUndergraduate · Chapter 645
Logistic differential equations · Capacity K=6; Growth rate r=1.6
Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=6; Growth rate r=1.6.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 646
Euler stability · Decay rate a=1.25; Step size h=0.1
Use n=6 updates, corresponding to time t=nh=0.6. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.25; Step size h=0.1.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 647
Newton root iteration · Number whose root is sought A=10; 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=10; Positive starting guess b=7.5.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 648
Orthogonal projections · Vector horizontal a=2.5; 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=2.5; Vector vertical b=-1.
Question · Worked solution · Animated example
Open practice chapterUndergraduate · Chapter 649
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 chapterUndergraduate · Chapter 650
Matrix conditioning · Small diagonal entry a=0.45; 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=0.45; Final data error b=0.3.
Question · Worked solution · Animated example
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