Math 101 ~ Single Variable Calculus II

Class cancelled due to missing students!

Review: Math 100 Essentials

Review: Math 100 Essentials

7.1 Log and exponential functions

7.2 & 7.3 Derivatives and integrals of log and exponential functions
7.4 Graphs and applications of log and exponential functions.

7.5 L'Hopital's rule

7.7 & 7.8 Inverse trigonometric functions & Hyperbolic functions

8.2 Integration by parts

Summary

• Compare the two integrals ∫ x e^x2 dx and ∫ x e^2x dx. Typographically they differ only by the position of the digit 2 in the exponent, but they require very different integration approaches. The first can be done "By Substitution" (let u = x²), but the second requires integration by parts (let u = x, dv = e^2x).
• Derivation (from product rule)
• Formula: ∫ u dv = u v - ∫ v du
• Examples: 8.2 - 2, 6, 8
• Type: ∫ xⁿ cyclical-fn dx

  Let u = xⁿ. This will reduce its power by one in the new integral. Thus in questions 2 and 6 we only had to apply integration by parts once to eliminate the x term, but in 8 we had to apply it twice.

• Example: 8.2 - 10

  This time letting u = x didn't work. The reason is that ln x isn't a cyclical function. Instead letting u = ln x works because then du is dx/x.

• Example: ∫ ln x dx by using integration by parts and letting u = ln x and dv = dx.
• Example: 8.2 - 19 This is an example of a circular integral. We apply integration by parts twice after which the original integral has reappeared on the RHS! So we solve for it algebraically.
• Type: Circular integral

  Apply integration by parts repeatedly until the original integral appears on the RHS (usually two applications will suffice), then solve for it algebraically.

8.3 Integration of Trignonometric Functions

Summary

• ∫ n^m x dx, ∫ cos^m x dx

  For low powers substituting using the double angle identities can be faster than using the reduction formulae from 8.2
  ∫ sin x dx = -cos x + C
  ∫ sin²x dx Substitute using double angle identity to get linear cos term.
  ∫ sin³x dx Separate one sin x term to be differential and substitute pythagorean identity for remaining sin² term; integrate by substitution.
  ∫ sin⁴x dx Rewrite as sin²x sin²x; substitute double angle identity for both sin²s; expand; substitute for remaining sin² term; integrate individual terms.
  ∫ sin⁵x dx For this and higher powers use the reduction formula.
  

• Deriving reduction formulae for powers of sin and cos.
• ∫ sin^m x cos x dx, ∫ cos^m x sin x dx

  When one of the trig functions is to the power 1, integrate by substitution

• ∫ sin^m x cosⁿ x dx

  If one of the powers is odd, separate one of that function to use as the differential, substitute for the remainder using the pythagorean identity to produce a polynomial in that trig function.

  If both powers are even, substitute for both terms using the double angle identities.

  These rules are nicely summarized in Table 8.3.1 on page 536

• ∫ tan^m x secⁿ x dx

  The rationale remains the same: separate out a part of the expression to serve as the differential, the use the pythagorean identity to simplify what's left. Summarized in Table 8.3.2 on page 539.

8.4 Trigonometric Substitutions

Summary

• Recall: ∫_-1¹ √1-x² dx
  We were able to evaluate this definite integral by realizing that it is asking for the area of a semicircle, and using a geometric formula to evaluate it.
• Now what about ∫√1-x² dx?
  The difficulty this time is that it is an indefinite integral so we can't resort to geometry, and we can't use substitution or integration by parts because the integrand doesn't have two parts to it. So in desperation we try a trig substitution by letting x = sin θ this allows us to use the pythagorean identity to get rid of the radical, evaluate the resulting trig integral, and back substitute (using a triangle) to get the solution in terms of the original variable. Whew.
• The same method can be used for other integrals involving radicals.
• If the radical has form √1 - x² let x = sin θ¸ Example: 8.4-4
• If the radical has form √x² - a let x = sec θ¸ Example: 8.4-8
• If the radical has form &radic1 + x² let x = tan θ¸ Example: 8.4-6
• The last substitution is also sometimes used when there is a squared expression without a radical. Example: 8.4-16

Lecture: 8.5 Partial fraction expansion

Summary

• Our final transformation technique is applicable to rational expressions. It works by expanding a complex rational expression into a sum of simpler rational expressions, each of which yields to one of the integration techniques we have already covered.
  1. If the expression is not proper, divide the denominator into the numerator. The result will be a polynomial and a rational remainder.
  2. Factor the denominator into a set of linear and quadratic factors.
  3. Write out the partial fraction decomposition of the rational expression:
     • For each linear factor in the denominator introduce a term with that denoninator and a constant numerator, i.e. A/(x+c).
     • For each quadratic factor in the denominator introduce a term with that denoninator and a linear term as numerator, i.e. (Ax+B)/(ax²+bx+c).
  4. Find the values of the unknown parameters, A, B, C, ...
  5. Integrate the decomposition.

8.6 Integrating expressions with rational powers of x
8.8 Improper integrals

Summary

• For integrands containing multiple rational powers of x, i.e. x^m/n, try substituting u = x^1/n where n is lowest common denominator of the rational powers.
• For integrands containing radicals where x is not squared, try letting u = contents of the radical.
• When one of the limits of integration of a definite integral is infinity or a point where the function is undefined, substitute a parameter (k in class) for the limit, evaluating the integral, and then taking the limit of the result as k approaches the value.
• Note well our poor intuitions about infinity.

8.7 Numerical Integration

Summary

• We turn to numerical methods for computing definite integrals when they are too difficult to evaluate symbolically.
• Endpoint approximations (inscribed, circumscribed)
• Trapezoidal approximation
• Midpoint Approximation
• Simpson's rule

9.1 Differential Equations

9.1 Differential Equations and Mixing Problems

9.3 Differential Equations and Exponential Growth

Midterm

10.1 & 10.2 Infinite Sequences

• Notations.
• Graphs.
• Convergence.
• Monotonicity: Difference, ratio and derivative tests to assess monotonicity (monotone or strictly monotone and increasing/non-decreasing/non-increasing/decreasing).

10.3 Infinite series convergence

• Definition of convergence: lim S_n
• Trick: Subtracting fraction of S_n from itself to get closed form expression.
• Trick: doing a partial fraction expansion can yield a telescoping or light sabre series.
• Geometric series.
• Harmonic series diverges.

10.4 Three more convergence tests

• Divergence test. NB a one-way test.
• Integral test. An existence theorem.
• Hyperharmonic series/P-series
• Handy rule: deleting a finite number of finite terms from an infinite series does not affect its convergence.

10.5 Ratio, Root and Comparison Tests

Class cancelled.

10.6 Alternating series; Conditional convergence

10.8 Power series

10.7 Maclaurin and Taylor series and polynomials

10.9 Remainder Estimation Theorem (Convergence of Taylor Series)

Summary

• Steps in approximating a value y using a series approximation:

  1. Choose a function that takes the value y, i.e. choose f such that f(x)=y
  2. Choose a point a near x at which to expand the series (choose a so that the derivatives of f at a are easy to find).
  3. Expand the series about a, i.e. calculate f(a), f'(a), f''(a), etc. and write the terms in the resulting series.
  4. Use the Remainder formula to estimate how many terms of the series will be needed to get the desired accuracy. (Note the key step here is to bound the unknown derivative f^(n+1)(c) ).
  5. Use the number of terms indicated in 4. to calculate f(x) = y.

11.1 Polar Coordinates

11.3 Area in Polar Coordinates

Parametric Equations

Final Exam