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Name: Student ID: Section: Instructor:
Math 112 (Calculus I)
Final Exam Form A
December 15, 2009
Instructions:
- Work on scratch paper will not be graded.
- In the multiple choice and short answer sections, only the answer will be graded. There will be no partial credit.
- For questions 10 to 20, partial credit will be given if you show all your work in the space provided. Full credit will be given only if the necessary work is shown justifying your answer. Please write neatly.
- Should you have need for more space than is allotted to answer a question, use the back of the page the problem is on and indicate this fact.
- Simplify your answers. Expressions such as ln(1), e^0 , sin(π/2), etc. must be simplified for full credit.
- Calculators are not allowed.
For Instructor use only.
Possible Earned
MC 24
9 11
10 8
11 6
12 4
13 3
14 4
Sub 60
Possible Earned
15 10
16 6
17 6
18 6
19 6
20 6
Sub 40
Total 100
Multiple Choice. Fill in the answer to each problem on your computer-scored answer sheet. Make sure your name, section and instructor are on that sheet.
- Approximate
1
x^4 dx using a Left Hand sum with 2 subintervals (n=2).
(a) 82 (b) 164 (c) 81 (d) 162 (e) 624 (f) 625 (g) None of these
- Find the area under the function f (x) = 3
x from x = 1 to x = 8.
(a)
(b)
(c) 12 (d) 15 (e)
(f) None of these
- Given the limit statement lim x→ 1 (2x − 3) = −1 pick the largest δ that works with the definition of the limit if = 0.06. (a) 0.001 (b) 0.005 (c) 0.01 (d) 0.02 (e) 0.03 (f) No such δ exists
- Which of the following is an inflection point of f (x) = x x^2 + 1
(a) 1 (b) − 1 (c) 2 (d) − 2 (e)
2 (f) −
2 (g) − 3 (h)
- Given x ln y − y ln x = e^2 − 2 e, find dy dx at the point (e^2 , e).
(a) 0 (b) e (c) e^2 (d) 1 − e e^2 (e) 1 − e e^2 − 2 e (f) e^2 − 2 e (g) e − 1 e^2
- Which of the following are x-values for which f (x) = sin(x) − x has a local maximum?
(a) − 2 π (b) −π (c) 0 (d) π (e) 2 π (f) More than one of these (g) None of these
- Which of the following functions has a discontinuous first derivative?
(a) sinh(x) (b) x^1 /^3 (c) tan−^1 (x) (d) x 1 + x^2 (e) ln(x^2 + 1)
(f) All of the first derivatives of these functions are continuous
d dx
∫ (^2) x
1
1 + t^3 dt =
a)
1 + (2x)^3 −
2 b) 2
1 + (2x)^3 −
2 c)
1 + x^3 −
2]
d) 2
1 + x^3 −
2 e)
1 + (2x)^3 f) 2
1 + (2x)^3
g)
1 + x^3 h) 2
1 + x^3
Free response: Write your answer in the space provided.
- (8 points)
(a) If f (x) =
x
, use the definition of a derivative to set up a limit to find f ′(x).
(b) Find f ′(x) by evaluating the limit. (No points will be awarded if differentiation rules are used.)
- (6 points) Find the dimension of the largest rectangle that can be inscribed between the curve y = 4 − x^2 and the x-axis.
- (4 points) lim x→ 0 ln(x) sin(x)
- (3 points) d dx
ln
xex^ − sin x x
- (4 points)
x x^2 + 4 dx
- (6 points) A certain element has a half life of 20 years. How many years will it take until only
10% of the element remains?
Note: ln
≈ −.7 and ln
≈ − 2 .3. You can either leave
your answer in terms of logs or give a numerical answer using these approximations.
- (6 points) The equation of the tangent line to the curve y =
x^2
at
- (6 points) Use linear approximation to estimate
- (6 points) A pump is blowing up a spherical balloon with a pump rate of 10cm^3 /sec. How fast
is the diameter of the balloon growing when the balloon has a 5cm radius?
Volume of a sphere is given by
πr^3.
- (6 points) A particle is moving with the given data. Find the position function of the particle.
a(t) = sin(t) + 3 cos(t), s(0) = 0, v(0) = 2.
END OF EXAM
