Calculus I :: Exam 1 Review

I plan to have around 10 questions on the exam with at least one question from each section. You should be able to state any definition and any named theorem. You should be able to work any homework problem which has been assigned. Here is a list of important ideas:

• Average velocity
• Estimating instantaneous velocity
• Secant lines/Tangent line
• Graphing lines
• Rates of change, average and instantaneous
• Numerical limits/Tables
• Graphical limits
• Intuitive definition of limit, and proper notation
• Theorem on page 65 (2.2 Theorem)
• Left and right hand limits
• Infinite limits
• Vertical and horizontal asymptotes
• Theorem on page 72 (Basic Limit Laws), and how to use it
• Intuitive and precise definition of continuity
• Types of discontinuity and their definitions (removable, jump, infinite)
• One sided limits
• Piecewise functions
• Basic laws of continuity
• Continuity of polynomial/ration/other basic functions
• Continuity of inverse and composite functions
• Ideas for evaluating limits algebraically
• When a function is in indeterminate form
• The Squeeze Theorem and Theorems on page 90
• Infinite limits vs. limits at infinity
• Theorems for evaluating limits at infinity
• The Intermediate Value Theorem and its applications
• The bisection method for estimating roots

1. A stone is tossed vertically into the air from ground level with an initial velocity of 15 m/s. Its height at time \(t\) is \(h(t) = 15t-4.9t^2\) m.
   1. Compute the stone's average velocity over the time intervals [1,1.01], [1,1.001], [.99,1], [.999,1].
   2. Estimate the stone's instantaneous velocity at \(t=1\).

2. Use the Basic Limit Laws and the facts that for constants \(c\) and \(k\),

   • \(\displaystyle \lim_{x\to c} k = k\)
   • \(\displaystyle \lim_{x\to c} x =c\)

   to evaluate \(\lim_{x\to2} \sqrt{x^3+5x+7}.\)

3. Let \(g(x)\) be the function \[g(x) = \begin{cases} \ln(-x-3) & \textrm{if } x < -3 \\ \left| 1-x \right| & \textrm{if } -3 \leq x < 2\\ 4 & \textrm{if } x = 2 \\ 2 \sqrt{2x} & \textrm{if } 2 < x \leq 8 \\ 8 & \textrm{if } x > 8 \\ \end{cases} \] Determine whether the following are true or false.
   1. \(\displaystyle \lim_{x\to -3^+} g(x) = g(-3)\).
   2. \(g(x)\) has a removable discontinuity at \(x=-3\).
   3. \(\displaystyle \lim_{x\to 2^-} g(x) = g(2)\).
   4. \(g(8) = 8\).
   5. \(\displaystyle \lim_{x\to 8} g(x)\) exists.
   6. \(g(x)\) is continuous at \(x=8\).
   7. \(g(x)\) has a jump discontinuity at \(x=8\).
4. State the Squeeze Theorem, then us it to show that \(\displaystyle \lim_{x\to 0} x \sin \frac{1}{x} = 0\).
5. Find all vertical asymptotes of \(\displaystyle \frac{x^2-4}{x^2-2x}\).
6. Find all horizontal asymptotes of \(\displaystyle \frac{2x^2+11x+12}{2x^2+x-3}\).
7. Show that \(\displaystyle \cos x + \frac{1}{2} = 2^x\) has at least one real root in the interval \([0,\pi]\).
8. Find the limit if it exists. If necessary, state whether the limit is \(\infty\), \(-\infty\), or does not exist. Show your work.
   • \(\displaystyle \lim_{x\to 4} \frac{16-x^2}{x-4}\).
   • \(\displaystyle \lim_{x\to \infty} \sqrt{36x^2-x} -6x\).
9. Prove rigorously (using the formal definition of limit) that \(\displaystyle \lim_{x \to 2} 4x-1 = 7\).
10. State the Intermediate Value Theorem.

Solutions to the most of the above are here, the may be in a different order, you can ignore question 1 in the solutions pdf. There is a small error in the solutions, bonus point if you find it.