A. Armstrong
MA 141- Fall 2012 - Test 4 Review
Area, Surface Area, and Volume Formulas
• Area
– Square: A = x2
– Rectangle: A = lw
– Triangle: A = 12 bh
– Circle: A = πr2
• Surface Area
– Cube: SA = 6x2
– Box with Square Base (closed top): SA = 2x2 + 4xh
– Sphere: SA = 4πr2
– Cylinder: SA = 2πr2 + 2πrh
• Volume
– Cube: V = x3
– Box: V = lwh
– Sphere: V = 34 πr3
– Cone: V = 31 πr2 h
– Cylinder: V = πr2 h
1. True or False:
(a) When maximizing a function, we solve the objective equation for one variable and plug into
the constraint equation.
(b) The most general antiderivative of f (x) = x−2 is F (x) =
(c) If
f0
Z
is continuous on [1, 3], then
3
−1
+ C.
x
f 0 (v)dv = f (1) − f (3).
1
Z
(d)
2
(x − x3 )dx represents the area under the curve y = x − x3 from 0 to 2.
0
(e) The right hand approximation for finding Riemann sums is always an overestimation.
Z b
Z c
Z b
(f)
f (x)dx =
f (x)dx −
f (x)dx.
a
a
c
2. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing
north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?
1
3. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter
of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 ft,
find the dimensions of the window so that the greatest possible amount of light is admitted.
4. Use Newton’s method to find the root of the equation,
1
= 1 + x3 , that is greater than 0. (Hint:
x
Start with x1 = 1.)
3
5. Find f (t) given f 00 (t) = √ with f (4) = 20 and f 0 (4) = 7.
t
6. A particle moves with acceleration a(t) = 5 + 4t − 2t2 (in m/s2 ). Its initial velocity is 3 m/s and
its initial position is 10 m. Find the position after t seconds.
7. Estimate the area under the curve, f (x) = 3 − 21 x2 , for 2 ≤ x ≤ 14 with 6 subintervals using
(a) left endpoints.
(b) right endpoints.
(c) midpoints.
(d) Which type of approximation is most accurate?
8. Evaluate
Z 1 the integrals.
2
1
(a)
(1 + u4 + u9 )du
2
5
0
Z √3/2
6
√
dt
(b)
1 − t2
1/2
Z π/3
sin θ + sin θ tan2 θ
(c)
dθ
sec2 θ
0
9. Water flows from the bottom of a storage tank at a rate of r(t) = 200 − 4t liters per minute where
0 ≤ t ≤ 50. Find the amount of water that flows from the tank during the first 10 minutes.
Z
10. Find
sec t(sec t + tan t)dt.
11. Evaluate
Z 1 the following.
d tan−1 x
(a)
(e
)dx
0 dx
Z 1
d
−1
(b)
(etan x )dx
dx 0
Z x
d
−1
(c)
((etan t )dt
dx 0
Z xp
Z
2
12. If f (x) =
1 + t dt and g(y) =
0
3
y
f (x), find g 00 (3).