MATH 135 – Calculus 1 Sample Questions for Exam 2 October 10, 2016

1. An object moves along a straight line path with position given by x(t) = 4t^{2} + t − 7, (t in seconds, x in feet).

(a) What is the average velocity of the object over the interval [0, 5] of t-values?

(b) Fill in the following table with average velocities computed over the indicated intervals. Using this information, estimate the instantaneous velocity at t = 2.

interval [2, 2.5] [2, 2.05] [2, 2.005] [2, 2.0005]
ave.vel.

(c) Construct a similar table for intervals ending at t = 2 and repeat the calculations in the previous part. If you estimate the instantaneous velocity at t = 2 using this new information, does your result agree with what you did before (it should!)

2. (a) What is the slope of the secant line to the graph y = x^{3} + 1 through the points with x = 1 and x = 2?

(b) What is the slope of the secant line to the graph y = x^{3} + 1 through the points with x = 1 and x = 1 + h for a general h?

(c) The slope of the tangent line to y = x^{3} + 1 at x = 1 would be obtained from what limit?

(d) Estimate the limit in the previous part numerically (as in the first question).

(e) Evaluate the limit exactly using our algebraic techniques.

3. The graph of a function f is shown below with several points marked. Find all the marked points at which the following are true, and give explanations for your answers.

A  B  C D E  F  G

(a) f has an infinite discontinuity

(b) f has a jump discontinuity (and: If you had a vertical axis with actual y-values to use, how would you find the one-sided limits at each such point?)

(c) f has a removable discontinuity

(d) f is continuous

4. Compute the indicated limits. Show all work for full credit.

(a) lim_{x→1} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}
(b) lim_{x→2} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}
(c) lim_{x→∞} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}

(d) lim_{x→2} \frac{\frac{1}{x^{2}} − \frac{1}{4}}{x − 2}
(e) lim_{t→0} \frac{sin(6t)}{sin(7t)}
(f) lim_{h→0} \frac{\sqrt{h + 9} − \sqrt{9}}{h}

5. Suppose you know each of the following conditions. What can you say about lim_{x→c} f(x) for the indicated c? Why?

(a) x^{2} + x ≤ f(x) ≤ x^{3} + 3 for all real x, at c = 0.

(b) −x^{2} + 2x ≤ f(x) ≤ x^{4} − 4x^{3} + 6x^{2} − 4x + 2 for all real x, at c = 1

(c) f(x) = x sin (\frac{1}{x}) for all real x ≠ 0, at c = 0

6. (a) What is the definition of the derivative of a function f at x = a in its domain?

(b) Using the definition (not the shortcut rules), find f′(x) for f(x) = 3x^{3} − 2x^{2} + 1 at a general x = a.

(c) Using the definition (not the shortcut rules), find f′(x) for f(x) = x^{1/2} at a general x = a > 0.

(d) Using the definition (not the shortcut rules), find f′(x) for f(x) = \frac{1}{x^{2}} at a general x = a ≠ 0.