MATH 141 Midterm: MATH141 South Carolina 141 96 1 nospace
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Exam 1, Math 141, 1996
PRINT Your Name: Section:
There are 8 problems on 3 pages. Problems 1 and 2 and are worth 20 points
each. The other problems are worth 10 points each. In problem 3 you MUST use
the definition of the derivative; in the other problems you may use any legitimate
derivative rule. SHOW your work. CIRCLE your answer.
NO CALCULATORS!
1. (The penalty for each mistake is five points.) Let
f(x)=
4−x
2if x<0,
xif 0 ≤x≤1, and
2−xif 1 <x.
(a) Graph y=f(x).
(b) Fill in the blanks:
f(0) = lim
x→0+f(x)= lim
x→0−
f(x)= lim
x→0f(x)=
f(1) = lim
x→1+f(x)= lim
x→1−
f(x)= lim
x→1f(x)=
f(2) = lim
x→2+f(x)= lim
x→2−
f(x)= lim
x→2f(x)=
(c) Where is f(x) continuous?
(d) Where is f(x) differentiable?
2. Compute the following limits:
(a) lim
x→3+
x2−9
x−3
(c) lim
x→3+
x−3
x2−9
(c) lim
x→3+
x2−9
x+3
(d) lim
x→3+
x+3
x
2−9
3. Use the DEFINITION of the DERIVATIVE to find the derivative of
f(x)=4
√
x−3.
4. Graph y=3+sinx.
5. Find the equation of the line tangent to f(x)=9x
10 +8xat x=−1.
6. Let f(x)=x
2cos x.Findf
′
(x).
7. Let f(x)= x
3+9x
sin x.Findf
′
(x).
8. Let f(x)=9x
3+9
x+4
√
x+16. Find f′
(x).
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