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A math test for the course math 1131, covering topics such as limits, discontinuities, and tangent lines. The test includes various problems, including evaluating limits, identifying discontinuities, and finding equations for tangent lines. Students are required to show their work and justify their answers.
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Test 2a, Math 1131 Name: _________________________ Sept. 30, 2004, Dr. Howard
Please show all work and justify all answers.
1. Evaluate the limits, if they exist.
a. lim x → 3 ( 4 x^2 − 5 x + 7 )^ b. lim x → 5 +
| x − 5 | x − 5
c. lim x → 5
| x − 5 | x − 5
d. lim x →∞ cos( e −^ x^ )
e. lim x → 4
x − 2 x − 4
2. The graph of y = f ( x ) appears below. Use it to determine lim x → 1
f ( x ) if it exists.
-2 -1 0 1 2
0
x
y
x
y
30 Sept. 2004, Dr. Howard
3. Give an example (or a graph) of a function defined on the interval [−4, 4] and having a jump discontinuity at x = 1. 4. Identify all vertical asymptotes, horizontal asymptotes, and removable discontinuities of the
rational function r ( x ) = 3 x
(^2) − 5 x − 2 9 x^2 − 19 x + 2
5. Prove that the equation arcsin( x / 4 ) = 1 + sin x has at least one real root, then use your calculator to find an interval of length 0.01 that contains a root.
30 Sept. 2004, Dr. Howard
8. Suppose that a tennis ball was shot into the air and that its height versus time is given by the data in the table below. Estimate the instantaneous velocity of the tennis ball 2 seconds after it was fired.
Time t (seconds) 0 1 2 3 4 5 Height (feet) 8.0 140.2 240.2 308.0 343.6 347.
9. Tangent Line Concept. Explain in words and pictures the process that we followed in class to obtain the slope of the line tangent to the curve y = f ( x ) at x = a using secant lines.
