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ins5-2h22-28 In Problems ??, ﬁnd the area of the regions between the curve
and the horizontal axis 5-2h22
5-2h23
5-2h24
5-2h25
5-2h26
5-2h27
5-2h28
5-2h29 2 f (x) 1
Math for Econ II, Written Assignment 8 (28 points)
3
x
22. Under y = 6x − 2 for 5 ≤ x ≤ 10.
Due Friday, April 17
−1
23. Please the curve y = solutions ≤ t ≤ π/2.
Under write neat cos t for 0 for the problems below. Show all your work. If you only write the answer with no work,
5-2h32fig −2
24. you will = ln be given x ≤ 4.
Under y not x for 1 ≤ any credit.
2
4
6
8
10 25.
26.
27.
28.
29. Under y = 2 cos(t/10) for 1 ≤ t ≤ 2.
√
• Write your name and recitation section number. Figure 5.39: Graph consists of a semicircle and
Under the curve y = cos x for 0 ≤ x ≤ 2.
line segments
Under the curve y =homework above the x-axis.
• Staple your 7 − x2 and if you have multiple pages!
Above the curve y = x4 − 8 and below the x-axis.
1. (2 pts total; 1 pt each) Use the ﬁgure below to ﬁnd the values of
Use Figure ?? to ﬁnd the values of
5-2h33 33. (a) Graph f (x) = x(x + 2)(x − 1).
b
c
f bf
(b)
f (x) dx
(a)
(b) Find the total area between the graph and the x-axis
(a)(x) dx(x) dx
a
b
c
c
c
between x = −2 and x = 1.
(c)
f (x)c |f (x)| dx (d)
dx
|f (x)| dx
a
a
1
(b) a
(c) Find −2 f (x) dx and interpret it in terms of areas.
f (x) √ b 5-2h29fig c 34. Compute the deﬁnite integral
the result in terms of areas. 5-2h35 35. Without computation, decide if 0 e−x sin x dx is positive or negative. [Hint: Sketch e−x sin x.] 5-2h36 ©
a 4 5-2h34 Area = 13 36. Estimate x sArea = 2 ! cos x dx and interpret 2π (a) Figure 5.36 0 1
0 2 e−x dx using n = 5 rectangles to form a Left-hand sum (b) Right-hand sum 5.4 THEOREMS ABOUT DEFINITE INTEGRALS 311 5-2h37 37. (a) On a sketch of y
0
graph of f in FigureFigure ??, estimate:
following
f (x)dx = 4 and ??, arrange the ins5-4h47-48 In Problems ??, evaluate = ln x, represent the left Riemann
5-4h43 43. Using the
5-2h30 30. Given
2
or
sum with quantitiesexpression, if ln x dx. Write least to greatest.
n = 2the in increasingpossible,from
approximating
−2
2. (4 pts) Using the graph of f in Figure, arrange the following
order,
quantities2in increasing order, from least to greatest.
say what out the terms in the sum, but do not1evaluate it.
additional information is needed, given that
2
2
2
3
2
4
(x)dx
f
(a) 1 0 f 1
2
i) (x) f (x) dx ii) (b) (x)(x)(x)dx
iv) = 12.
(b) On another sketch, − 1 f (x) dx vi) The number 0
f total
(ii)
f
(i)(c) 0 The 0 dx shaded area. 1 f1 −2dxdx iii) 0 f (x) dx g(x) dx 2 f (x) dx v) represent the right Riemann sum
−4
2
2
3
with n = 2 approximating 1 ln x dx. Write out the
(iii)
f (x) dx
(iv)
f (x) dx
4
4
0
2
f (x)
terms in the sum, but do not evaluate it.
2
5-4h48 48.
20 viii) The 0 5-4h47 47.
g(x) dx
g(−x) dx
(v) − vii) (x) dx
f The number (vi) The numbernumber −10
(c) Which sum is an overestimate? Which sum is an un1
2
0
−4
(vii) The number 20
(viii) The number −10
derestimate?
10 −2 5-2h30fig 5-4h43fig −10 −2 1 x f (x) 2
2 5-2h38 ins5-4h49-52 3 Figure 5.37 x ! 38. (a) Draw the rectangles that give the left-hand sum apπ
proximation to 0 expression if possible,
In Problems ??, evaluate the sin x dx with n = 2. or say
7
0
what extra information is needed, given dx. (x) dx = 25.
(b) Repeat part (a) for −π sin x 0 f
(c) From your answers to parts (a) and (b), what is
sum approximation to
7 the value of the left-hand 3.5
π
5-4h50 50.
49.
f−π sin x dx with n = 4?
(x) dx
f (x) dx 0
Figure 5.71
5-4h49
31. (a) Using Figure ??, ﬁnd −3 f (x) dx.
0
0
(b) If the area of the shaded region is A, estimate
5-4h44 44. (a) Using Figures ?? and ??, ﬁnd the average value on
5-2h39 39. (a) Use a calculator or computer to ﬁnd 6 (x2 + 1) dx.
4
−1
4
4
5
7
dx.
3. ≤−3 f (x)of
dx =
f (x) dx = value as the ﬁnd 1 f0(x) dx,
5.
0 (3 pts) Suppose f is even, −2 f (x) 5-4h51 3, and 2Represent this5-4h52Either area under a curve. or show there is not
x≤2
51.
f (x + 2) dx 6 2
52.
(f (x) + 2) dx
4
(i) enough information to ﬁnd 1 f (x) dx.
f (x)
(ii) g(x)
(iii) f (x)·g(x)
(b) Estimate 0 (x + 1) dx using a left-hand sum with
1
−2
0
f (x)
n = 3. Represent this sum graphically on a sketch
(b) Is the following statement true? Explain your an4
4. (5 pts) Evaluate the following integrals:x
of f (x) = x2 + 1. Is this sum an overestimate or
swer.
−4 −3 −2 −1
1
2
3
55-4h53 53. (a) Sketch a graph of f (x) = sin(x2 ) and mark on it
underestimate of the true value√
found in part (a)?
√
√
5-2h31fig
Average(f )−1If 1 (f (x) − 2g(x)) dxg) 6 and 1 (2f the points x = √π,= 2π,ﬁnd 1 (f (x) − g(x)) dx.
·
6 dx
(x) + 2g(x))(x2 +1)9, using 0 right-hand sum with
(a) (3 pts) Average(g) = Average(f · =
0 (c) Estimate
0
dx 3π, a 4π.
0
3
(b) Use your graph to decide which of yourfour numbers
1
n = 3. Represent this sum on the sketch. Is this
Figure19
5.38
f (x)
g(x)
3
(b) (2 pts)
(x − 4x + x) dx
√
sum an overestimate or underestimate?
nπ
−3
sin(x2 ) dx n = 1, 2, 3, 4 5-2h31 5-4h44figa x Figure 5.72
5-4h45 5-4h46 x 0
5-4h44figb
5. (2 pts) Explain what is wrong with2the following calculation of the area under the curve
1
2
1 Figure 5.73 1 Area = √1 b −x2 /2
e from x = −1 to x = 1: is largest. Which is smallest? How many of the num- 1
1 1
1 1
2
bers are
dx = − 3 positive? − = − .
=−
4
x
3x −1
3 3
3 −1
45. (a) Without computing any integrals, explain why the
ins5-4h54-56
average value of f (x) = sin x on [0, π] must be be6. (3 0.5 and 1.
tween pts) Suppose the area under the curve
x = 0 to x = b.
(b) Compute this average. Solve for a in terms of b.
46. Figure ?? shows the standard normal distribution from
statistics, which is given by
2
1
√ e−x /2 .
2π
Statistics books often contain tables such as the following, which show the area under the curve from 0 to b for
various values of b. ' 1
x4 For Problems ??, assuming F = f , mark the quantity on a x
ecopy of Figure0??. x = a is six times the area under the curve 2e2x from
from x = to
(in other words, write a =(some formula involving b)) F (x) ins5-4h54-56fig a b Figure 5.75
dx x 5 1
dx = ln(5). Now ﬁnd a fraction which approximates ln(5), by
x
1
5
1
using M4 (midpoint sum with 4 rectangles) to approximate
dx.
1 x
(The actual value of ln(5) is 1.6094 . . .. For fun, plug your approximation into a calculator and compare) 7. (4 pts) Use the Evaluation Theorem to show 6000
and the supply curve is given by P = Q + 10. Find
Q + 50
the equilibrium price and quantity, and compute the consumer and producer surplus. 8. (5 pts) Suppose the demand curve is given by P = Some extra practice (not to be handed in)
3 f (x) dx using R5 and L5 . 1. Estimate −2 −1 1 2 3 −5 −4 −3 −2 −3 −1 1 2 3 4 5 −2 2. Suppose h is a function such that h(1) = −2, h (1) = 3, h (1) = 4, h(2) = 6, h (2) = 5, h (2) = 13, and h is
2
continuous everywhere. Find 1 h (u) du. ...

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