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Instructor’s Name: Student’s Name: 1
Sample Final of Math 121, Fall 2005 Print your name on every page. There are 7 pages with 15 problems. Two detachable blank pages are provided at the back of this test for use as a scratch paper only, and any work left on this scratch paper will NOT be graded.
(54 = 18 × 3 points) Part I. Multiple-choice problems. Circle the correct answer. No partial credit possible.
- In three hours, the velocity of a car at each half hour was recorded as follows:
Time (hours) 0 .5 1 1.5 2 2.5 3 Velocity (mi/h) 25 45 50 55 40 45 37
What is the numerical estimate of the average velocity (in miles/hour) of the car over these three hours, obtained via the Simpson’s approximation S 6? (a) 130 (b) 133 (c) 136 (d) 137 (e) none of the above
The slope of the tangent line to the curve x^2 + x ln y + xy^2 = 5 + ln 2 at the point (1, 2) is (a) −^32 (b) −^43 (c) −^43 − 29 ln 2 (d) −^56 − 29 ln 2 (e) none of the above
If 600 cm^2 of material is available to make an open-top box with a square base, what is the largest possible volume of the box? (a) 1000
(b) 10
(c) 10
(d) 100
(e) none of the above
- Let f and g be functions with continuous derivatives f ′^ and g′, respectively, well defined on the real line, such that
1 f^ (x)^ dx^ = 4 and^
1 f^ (x)^ dx^ = 6. Furthermore we have the following data. x f (x) g (x) f ′^ (x) g′^ (x) 1 − 3 − 1 2 − 3 3 5 1 − 1 − 2 5 − 5 0 − 2 3
(i) (g ◦ f )′^ (3) = (a) − 3 (b) 2 (c) − 4 (d) 4 (e) none of the above
....................................................... (ii) If h (x) = f (x)^2 g (x^2 ), then h′^ (1) = (a) − 15 (b) − 48 (c) − 42 (d) 36 (e) none of the above .......................................................
(iii) limx→ 5
f (x) g (x) x^2 − 25
(a) 1 (b) − 1. 5 (c) ∞ (d) − 0. 6 (e) none of the above
....................................................... (iv) If H (x) =
∫ (^) ln x 0 f^ (t)
(^2) dt, then H′ (^) (e (^3) ) =
(a) (^253) (b) 8e^3 (c) 25 (d) 25e−^3 (e) none of the above
.......................................................
(v)
∫ (^) ln 3
0
etf
et^ + 2
dt =
(a) 4 (b) 0 (c) 2 (d) ∞ (e) none of the above
.......................................................
(vi)
1
xf ′^ (x) dx =
(a) 14 (b) 24 (c) 12 (d) 11 (e) none of the above
- If
− 3
f (x) dx = −1,
− 1
f (x) dx = 8, and
− 3
f (x) dx = 6, find
− 1
f (x) dx.
(a) 2 (b) − 1 (c) 3 (d) − 2 (e) none of the above
- Let F (x) =
∫ (^) x 2
0
e−f^ (t)dt for a differentiable function f on (−∞, ∞). Then F ′(x) =
(a) 2xe−f^ (x)f ′^ (x) (b) − 2 xe−f^ (x)f ′^ (x)
(c) 2xe−f^ (x
(d) − 2 xe−f^ (x
(e) none of the above
- A tank of the shape of a circular cone with its vertex pointing downward (and its top horizontal) is completely filled with water. Assume that the radius of its circular top is 1 m and its height (i.e. the distance from the vertex to the top) is 1 m. (i) What is the work, measured in newton-meter (i.e. joule or kg·m^2 /s^2 ), needed to pump all of the water out of this tank over its top? (Note that the density of water is 1000 kg/m^3 and the gravitational acceleration is 9.8 m/s^2 .) (a) 980012 π (b) 98003 π (c) 9800π (d) 98004 π (e) none of the above
....................................................... (ii) Let h (t) be the water level (i.e. the distance from the vertex to the water surface) in the tank t seconds after we start to pump the water out of this tank at the constant rate of 0.1 m^3 /s. How fast, in m/s, is the water level dropping at the moment when the water level is 0.4 m? (Give the correct rate of change in its absolute value, ignoring the ±-sign.) (a) 0.^ 277 8 π (b) 0.^ π^625 (c) 0. π^35 (d) 0.^ π^001 (e) none of the above
Part II. True-false Problems. Circle the correct answer, T (standing for ‘True’) or F (standing for ‘False’). No partial credit possible.
(8 = 4 × 2 points) 12. Determine whether each of the following statements is true or false.
(1) T F · · · · ·
1 f^ (x)^ dx^ =^
1 f^ (x)^ dx^ −^
9 f^ (x)^ dx^ for any continuous function^ f on (−∞, ∞).
(2) T F · · · · ·
2 f^ (x)^ dx <^ 0 for any^ positive^ (i.e.^ f^ (x)^ >^ 0 for all^ x) continuous function f on (−∞, ∞).
(3) T F · · · · · limh→ (^0 1) h
∫ (^) a+h a f^ (t
(^2) ) dt = 2af (a (^2) ) for any continuous function f on (−∞, ∞) and any a ∈ (−∞, ∞).
(4) T F · · · · · 16
3 f^ (t)^ dt^ ≤ |f^ (3)|^ +^ |f^ (9)|^ for any continuous function^ f^ on (−∞,^ ∞).
(14 = 7 × 2 points) 13. The graph of a continuous function f on the closed interval [− 6 , 7] is shown
in the following figure, where the arc is a semicircle. Let g(x) =
∫ (^) x
− 2
f (t) dt for − 6 ≤ x ≤ 7.
(1) T F · · · · · g (−6) < 0.
(2) T F · · · · · g (1) = g (3) = 6 − π.
(3) T F · · · · · g is not differentiable at x = −4, 0, 1, 3, 5.
(4) T F · · · · · g′′^ (4) = 4.
(5) T F · · · · · g has a local minimum at x = 72.
(6) T F · · · · · g is concave up on the interval (− 4 , 0).
(7) T F · · · · · g has an absolute maximum at x = 2 in the interval [− 6 , 7].
x
y
0
1
2
3
4
5
6
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7
(14 points) 15. The line y = −x + 4 and the parabola y = x^2 − 2 intersect at two point (− 3 , 7) and (2, 2), and bound (or enclose) a unique bounded region R. (i) Find the area A of the region R. (ii) Find the volume V , as a single concrete definite integral without actually computing the value, of the solid that has the region R (in the xy-plane) as its base such that each of its cross- sections perpendicular to the x-axis is a half-disk. (Note that this solid is not a solid of revolution.) (iii) Find the length ` of the whole boundary of the region R, as a single concrete definite integral without actually computing the value.
Blank scratch paper
