Quiz on 12 September 2018.

  1. Memorize the following definitions:
    • Proposition (Definition 1.2.1)
    • Prime Proposition (Definition 1.2.3)
    • Compound Proposition (Definition 1.2.5)
    • Paradox (Definition 1.2.7)
  2. Know the truth tables for the following:
    • Negation \neg
    • Conjunction \wedge
    • Disjunction \vee
    • Exclusive Disjunction \veebar
    • Implication \rightarrow
  3. Complete the following exercises in the textbook (Kohar).
    • Section 1.2—1.2.3, 1.2.4, 1.2.5
    • Section 1.3—1.3.1, 1.3.2, 1.3.5, 1.3.6, 1.3.10, 1.3.11

Here are some quick review videos for MAE 113.

Derivative Rules

Applications of the Derivative

Solutions

Tags: , , ,

I have posted assignment 8 here. The solutions for Section 11.3 are here. The solutions for Section 11.4 are here.

Tags: , ,

List of indefinite integrals
  • \int x^n \, \d x = \dfrac{1}{n + 1} x^{n + 1} + C (n \neq 1)
  • \int e^x \, \d x = e^x + C and the related \int e^{k x} \, \d x = \dfrac{1}{k} e^{k x} + C (k \neq 0)
  • \displaystyle\int \dfrac{1}{x} = \ln |x| + C
  • \int b^x \, \d x = \dfrac{b^x}{\ln b} + C (b \neq 1)

Any of the above forumale can be verified by differentiating the right hand side.  For example \int x^n \, \d x = \dfrac{1}{n + 1} x^{n + 1} + C (n \neq 1) because \dfrac{ \d \ }{\d x} \left( \dfrac{1}{n + 1} x^{n+1} + C \right) = x^n.  This gives us a way to check our answers.

We can apply the rules from the list of indefinite integrals above to more complicated functions.

Two Useful Properties of Integrals
  1. \int c f(x) \, \d x = c \int f(x) \, \d x

The integral of a constant times a function is equal to the constant times the integral of the function.  Basically, this means that we can pull the constant out in front of the integral.  This is similar to our constant multiple rule with derivatives.

2. \int \big( f(x) + g(x) \big) \, \d x = \int f(x) \, \d x + \int g(x) \, \d x

The integral of a sum of two functions is equal to the integral of the first function plus the integral of the second function. Basically, this means that the integral of the sum of two functions can be broken into two separate integrals for each function.

Examples.

  1. \int 3 x^2 \, \d x = 3 \int x^2 \, \d x by property 1.
  2. \int x + 3x^2 \, \d x = \int x \, \d x + \int 3x^2 \, \d x by property 2.
Examples Illustrating the Use of the Two Properties and the List of Indefinite Integrals
  1. Find \int (2 x^4 - 9x^3 - 2x + \pi) \, \d x.

    \begin{align*} \int (2 x^4 - 9x^3 - 2x + \pi) \, \d x &= \int 2x^4 \, \d x - \int 9x^3 \, \d x - \int 2x \, \d x + \int \pi \, \d x & \text{By Property 2}\\ &= 2 \int x^4 \, \d x - 9 \int x^3 \, \d x - \int 2x \, \d x + \int \pi \, \d x & \text{By Property 1}\\ &= \frac25 x^5 - \frac94 x^4 - x^2 + \pi x + C \end{align*}

2. Evaluate \displaystyle\int_1^4 \frac{1}{\sqrt{x}} \, \d x.

    \begin{align*} \int_1^4 \frac{1}{\sqrt{x}} \, \d x &= \int_1^4 x^{-\frac12} \, \d x\\ &= \left.\frac{1}{-\frac12 + 1} x^{-\frac12 + 1}\right|_1^4\\ &= \left.\frac{1}{\frac12} x^{\frac12}\right|_1^4\\ &= \left. 2x^{\frac12}\right|_1^4\\ &= \left. 2\sqrt{x}\right|_1^4\\ &= 2 \sqrt{4} - 2 \sqrt{1}\\ &= 2 \cdot 2 - 2\\ &= 4 - 2\\ &= 2 \end{align*}

3. Find \displaystyle\int_1^4 \frac{t^3 + \sqrt{t} - 4}{t} \, \d t.

    \begin{align*} \int_1^4 \frac{t^3 + \sqrt{t} - 4}{t} \, \d t &= \int_1^4 \left(t^2 + \frac{1}{\sqrt{t}} - \frac{4}{t}\right) \, \d t\\ &= \int_1^4 \left( t^2 + t^{-\frac12} - \frac{4}{t} \right) \, \d t\\ &= \left.\frac13 t^3 + 2 t^{\frac12} - 4 \ln t \right|_1^4\\ &= \frac13 (4)^3 + 2(4)^{\frac12} - 4 \ln 4 - \left[\frac13 (1)^3 + \underbrace{2(1)^{\frac12}}_2 - 4 \underbrace{\ln 1}_0 \right]\\ &= \frac{64}{3} + 4 - 4 \ln 2^2 - \frac13 - 2\\ &= \frac{64}{3} + 2 - 4 \ln 2^2 - \frac13 \\ &= \frac{63}{3} + \frac{6}{3} - 8 \ln 2\\ &= \frac{69}{3} - 8 \ln 2\\ &= 23 -  8 \ln 2 \end{align*}

Tags: , ,

If f(x) = b^x, then f'(x) = b^x \ln x.

Let f(x) = b^x. Using the definition of the derivative, we have

    \begin{align*} f'(x) &= \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}\\ &= \lim_{\Delta x \to 0} \frac{b^{x + \Delta x} - b^x}{\Delta x}\\ &= \lim_{\Delta x \to 0} \frac{b^x b^{\Delta x} - b^x}{\Delta x}\\ &= \lim_{\Delta x \to 0} \frac{b^x (b^{\Delta x} - 1)}{\Delta x}\\ &= \lim_{\Delta x \to 0} b^x \underbrace{\lim_{\Delta x \to 0} \frac{b^{\Delta x} - 1}{\Delta x}}_{\ln x}\\ &= b^x \ln b. \end{align*}

It would be nice if \displaystyle\lim_{\Delta x \to 0} \frac{b^{\Delta x} - 1}{\Delta x} = 1. Well, this occurs for a special value of b; we let b = e \approx 2.718.

This leads to the following:

If f(x) = e^x, then f'(x) = e^x.

Assignment 6 has been handed out. You can find a copy of it here.

Also, assignment 7 has been handed out. You can find a copy of it here.

Each of these assignments refer to the handouts that I gave in class. If you do not have the handouts, please come see me.

Tags: , ,

Do the questions from the large booklet that I handed out to you after the test.

You should have done Exercise 1 (all questions) already as instructed at the end of the test.

Exercise 2 (p. 352-3)

  1. b, d, f, h
  2. b, d, f, h
  3. a, c, e, g, j
  4. a, c, e, g, h

Exercise 3 (p. 354)

  1. a, b, d, g, h, i, j
  2. a, e, f
  3. b, c, d, e, f

Tags: , , ,

These are some highlights from a talk that E. O. Wilson gave at TED.

Keep your eyes lifted and your head turning. The search for knowledge is in our genes. It was put there by our distant ancestors who spread across the world, and it’s never going to be quenched. To understand and use it sanely, as a part of the civilization yet to evolve requires a vastly larger population of scientifically trained people like you. In education, medicine, law, diplomacy, government, business and the media that exist today.

Our political leaders need at least a modest degree of scientific literacy, which most badly lack today — no applause, please. It will be better for all if they prepare before entering office rather than learning on the job. Therefore you will do well to act on the side, no matter how far into the laboratory you may go, to serve as teachers during the span of your career.

I found out that in science and all its applications, what is crucial is not that technical ability, but it is imagination in all of its applications. The ability to form concepts with images of entities and processes pictured by intuition. I found out that advances in science rarely come upstream from an ability to stand at a blackboard and conjure images from unfolding mathematical propositions and equations. They are instead the products of downstream imagination leading to hard work, during which mathematical reasoning may or may not prove to be relevant. Ideas emerge when a part of the real or imagined world is studied for its own sake.

Tags: , , ,

Assignment 5 is posted here. Solutions were handed out in class.

Tags: , ,

The first test of MAE 113 will be on 13 February 2018.

No calculators, notes, or textbooks will be allowed.

You are responsible for all material

  • covered in class lectures
  • Assignments 1, 2, 3, and 4
  • will be asked to prove one of the following differentiation rules:

Tips:

  1. Review your two marked quizzes and identify your mistakes. You do not want to make these same mistakes again on the test.
  2. Practice. You should have completed all the assignments to feel confident in abilities.
  3. Drop-In Math Help Centre. It is open from 7 pm to 10 pm in G327. Tutors are available to help you. Take advantage of this opportunity!

Tags: ,

« Older entries § Newer entries »