MATH 2301-100 Spring 2022 Calculus I Recitation 3 Week 4

Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.

1. (10 points) Sketch the graph of a single function that is continuous except:
   1. at 1, where it has a removable discontinuity;
   2. at 2, where it has a jump discontinuity but is continuous from the right; and
   3. at 3, where it has an infinite discontinuity.
2. (25 points)
   1. State the Intermediate Value Theorem.
   2. Use the Intermediate Value Theorem to show that the equation \(x^2 =\cos(x)\) has a solution.
3. (25 points)
   1. Define "\(\displaystyle \lim_{x\rightarrow c} f(x) =\infty\)".
   2. Using the definition, prove that \(\displaystyle \lim_{x\rightarrow 3} \frac{5}{(x-3)^2} = \infty\).
4. (10 points) Compute \(\displaystyle \lim_{x\rightarrow -\infty} \frac{2x^2+3x^3-4}{5+7x^2+11x^3} =\)
5. (30 points) Let \(f(x)=\sqrt{x}-2\).
   1. State the definition of the derivative as a limit.
   2. Using this definition, compute \(f'(x)\).
   3. Find the equation for the tangent line at \(x=9\).
   4. Graph \(f(x)\) and the tangent line.

Last modified: Thu Jan 27 16:56:30 UTC 2022