Calculus 2, Fall 2016

Growth Mindset

"While dealing with classes and grades think to yourself 'Do I want to perform as well as I can?' If the answer is yes, start changing your mindset to a growth mindset. The first step in doing this is learn, learn, learn. Don't worry about looking good, and having the best grade, just learn. It's so much more important than to get good grades. The second step is to work with passion and dedication - effort is key. People with growth mindsets know that they have to work hard, and they enjoy it. They understand that effort is what ignites their ability and causes it to grow. The last rule is to embrace your mistakes and confront your deficiencies. They don't blame others for their failures, they take it and get help."

– Clara Mitchinson (one of my awesome Day1: Great Lakes students) summarizing Carol Dweck's video and an article about her work.

Priming

Sketch by hand the following together on the same set of axes. Be sure to clearly indicate which function is above the other on the intervals \((0,1)\) and \((1,\infty)\).

  1. \(\displaystyle y = \frac{1}{x}\) and \(\displaystyle y = \frac{1}{x^2}\).

  2. \(\displaystyle y = \frac{1}{x^2}\) and \(\displaystyle y = \frac{1}{x^{1/2}}\).

  3. \(\displaystyle y = e^x\) and \(y = \ln(x)\).

Concept questions

  1. Given \(\displaystyle \int_{-\infty}^0 e^x \, dx = 1\), what is \(\displaystyle \int_0^1 \ln(x) \, dx\)? Draw a picture.

  2. Given \(\displaystyle \int_1^\infty \frac{dx}{x^2} = 1\), what is \(\displaystyle \int_0^1 \frac{dx}{\sqrt{x}}\)? Draw a picture.

Concept questions

  1. What is the definition of \(\ln(x)\) in terms of an integral?

  2. Using the definition of \(\ln(x)\) in terms of an integral, explain why \(\ln(1/2)\) is negative.

Improper integrals

Let \(f\) be a continuous function. Improper integrals are defined as limits of proper definite integrals.

  1. \(\displaystyle \int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx\)

  2. \(\displaystyle \int_{-\infty}^b f(x) \, dx = \lim_{a \to -\infty} \int_a^b f(x) \, dx\)

  3. \(\displaystyle \int_{-\infty}^{\infty} f(x) \, dx = \lim_{a \to -\infty} \int_a^c f(x) \, dx + \lim_{b \to \infty} \int_c^b f(x) \, dx\)

Improper integrals

\[\int_1^{\infty} \frac{dx}{x^2} = \lim_{b \to \infty} \int_1^b \frac{dx}{x^2} = \lim_{b \to \infty} \bigg\lbrack \frac{-1}{x} \bigg\rbrack_1^b\]
\[ = \lim_{b \to \infty} \bigg( 1-\frac{1}{b} \bigg) = 1.\]

Improper integrals