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MATH 140 · Calculus· edited just now

Handwritten notes: MATH 140, lecture 7, the chain rule. If y = f(g(x)) then dy/dx = f′(g(x))·g′(x) — derivative of the outside times derivative of the inside. Three worked examples, a sketch of a tangent line, and a practice problem: y = √(1 + x³), find dy/dx.

Lecture 7 — Chain rule.pdf
MATH 140 · Lecture 712

The Chain Rule

d/dx f(g(x)) = f′(g(x)) · g′(x)

  • Outside: differentiate f, keep g(x) inside
  • Then multiply by the inside's derivative, g′(x)
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Chain rule, visually f( g(x) ) outsideinside
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Practice test · Eigenvalues· made from 2 notebooks
MATH 240 · Linear algebraPractice test

Eigenvalues & eigenvectors

10 questions · about 50 minutes · from Lectures 5–7 and your Eigenvalues notes

  1. Find the eigenvalues of A = 41234 pts

    det(A − λI) = (4 − λ)(3 − λ) − 2 = 0
    λ² − 7λ + 10 = 0 → λ = 2, 5

  2. For each eigenvalue in Question 1, find an eigenvector v with (A − λI)v = 0.4 pts

  3. True or false: if λ is an eigenvalue of A, then λ² is an eigenvalue of A². Explain why.3 pts

  4. Is A diagonalizable? Use your answers above.3 pts

MATH 240 · Linear algebraStudy guide

Eigenvalues — what to know

Concept map

Av = λv→det(A − λI) = 0→eigenvectors→diagonalize

Formulas

Av = λvdet(A − λI) = 0tr A = λ₁ + λ₂det A = λ₁λ₂

Likely questions

  • Find the eigenvalues of a 2 × 2 matrix
  • Eigenvectors from (A − λI)v = 0
  • When is a matrix diagonalizable?

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