**Unformatted text preview: **(c) Are the answers to parts (a) and (b) the same? Explain.
ins5-2h22-28 2 5 2
In Problems ??, ﬁnd the area of the regions between the curve
f (x)
Math for Econ II, Written Assignment 8 (28 points)
and the horizontal axis
1 5-2h22
5-2h23
5-2h24
5-2h25
5-2h26
5-2h27
5-2h28
5-2h29 Due Friday, November 7 x
22. Under y = 6x3 2 for 5 x 10.
Jankowski, Fall 2014
1
23. Under the curve y = cos t for 0 t
/2.
Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work,
2
5-2h32fig
24. you will = ln be given x 4.
Under y not x for 1
2
4
6
8
10
any credit.
25. Under y = 2 cos(t/10) for 1 t 2.
Figure 5.39: Graph consists of a semicircle and
• Write your name x for 0 x 2.
26. Under the curve y = cos and recitation section number.
line segments
27. Under the curve y =homework above the x-axis.
7 x2 and if you have multiple pages!
• Staple your
28. Above the curve y = x4 8 and below the x-axis.
1. (2 pts to ﬁnd the each)
29. Use Figure ??total; 1 ptvalues ofUse the ﬁgure below to ﬁnd the values of
5-2h33 33. (a) Graph f (x) = x(x + 2)(x
1).
Rb Rb
Rc
f
(b)
f (x) dx
(a)
(b) Find the total area between the graph and the x-axis
(a)(x) dx(x) dx
f
R a Rc
Rbc
c
between x = 2 and x = 1.
c
(c)
f
R1
a
(b)(x)adx (x)| dx (d) a |f (x)| dx
|f
(c) Find 2 f (x) dx and interpret it in terms of areas.
f (x) 5-2h34 Area = 13 5-2h35 a c b 5-2h29fig x
5-2h36 ! Area = 2 34. Compute the deﬁnite integral
the result in terms of areas. 0 36. Estimate R1
0 e x2 cos R2 35. Without computation, decide if
tive or negative. [Hint: Sketch e (a) Figure 5.36 R4 0
x x dx and interpret e x sin x dx is posisin 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
the ins5-4h47-48 In Problems ??, evaluate
following = ln x, represent the left Riemann
the expression, R 2 possible, or
if
= 2 approximating 1
2
2. (4in increasing order, from least to greatest. arrange whatfollowing quantities in increasingln x dx. that least to greatest.
pts) Using the graph of f in Figure,
the sum with n information is needed, order, Write
from
quantities2 R
say
additional
given
R
R2
R4
R2R
R2
R3
R2
out the terms in the sum, but do not evaluate it.
R 1 f1
f (x)dx
(a) i) (x)dx dx ii) (b) f (x)2dx iii)
2
f (x)
f (x) dx g(x) dx 2 f (x) dx v) represent thedx Riemann sum
iv) = 12.
f (x) right vi) The number 0
0
f (x) dx
(ii) 1 1 f (x) dx
(i)
4
0
0
1
(b) On another sketch,
R2
(c) 0 The total shaded area.
R2
R3
(iii)
f (x) dx
(iv)
f (x) dx
Z 4 with n = 2 approximating 1 ln x dx. Write out the
Z 4
0
2
R2
(x)
terms in the sum, but do not evaluate it.
5-4h48 48.
vii) (x) dx
20 viii)fThe 0 5-4h47 47.
g(x) dx
g( x) dx
(v)
f The number (vi) The numbernumber 10 R
5-4h43 43. Using the graph of f in Figure ??, arrange
5-2h30 30. Given 0 f (x)dx = 4 and Figure ??, estimate: (vii) 1 2
(viii) The number 20 The number
x 10
5-2h30fig 5-4h43fig 2
2 1
10 f (x) 2
2 3 x Figure 5.37 10 (c) Which sum is an overestimate? Which sum is an un4
derestimate? 0 5-2h38 38. (a) Draw
ins5-4h49-52
In Problems ??, ! the rectangles that give the left-hand sum apR
evaluate 0 expression if possible,
proximation to the sin x dx with n = 2. or say
R7
R0
what extra information is needed, given dx. (x) dx = 25.
(b) Repeat part (a) for
sin x 0 f (c) From your answers to parts (a) and (b), what is
Z 7 the value of the left-hand 3.5 approximation to
Z sum
R
5-4h50 50.
sin
f (x) dxx dx with n = 4?
f (x) dx R0
Figure 5.71
5-4h49 49.
31. (a) Using Figure ??, ﬁnd 3 f (x) dx.
0
0
R6
(b) R the area of the shaded region is A, estimate
If
5-4h44 44. (a) Using Figures ?? and ??, ﬁnd the average value on
5-2h39 39. (a) 4
R 1
4
computer7to R 4 (x + 1) dx.
Z 5 R Use a calculator or Either ﬁndﬁnd f0(x)2 dx, or show there is not
Z
3. (3 f (x)of
pts) dx.
Suppose f is even, 2 f (x) dx = 3, and 2 f (x) dx = 5.
0 x 2
1
3
5-4h51 51.
Represent
value 52.
R4
f (x + 2) dxthis5-4h52 as the area under a curve.
(f (x) + 2) dx
R6 2
(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
2
0
1
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
2
3
55-4h53 53. 1(a) Sketch a graph of f (x) = sin(x2 ) and mark on it
R11
R
underestimatedx the9, ﬁnd R 1found in part (a)? dx.
true value
5-2h31fig
(a) (3 pts) Average(g) = 2g(x)) dx =
(x) + 2g(x)) of
R6
Average(f ) · 1If 0 (f (x) Average(f · g) 6 and 0 (2f the points x = 2 ,= 2 , 3 , 0 (f (x) g(x))
4 .
(c) Estimate 0 (x +1) dx using a right-hand sum with
Z 3
(b) Use your graph to decide which of yourfour numbers
1
19
f (x)
n = 3. Represent this sum on the sketch. Is this
(b) (2 pts) Figure 5.38 4x3 + g(x)dx
(x
x)
Z n
sum an overestimate or underestimate?
3
sin(x2 ) dx n = 1, 2, 3, 4 1
5-2h31 5-4h44figa 5-4h45 5-4h46 x
x
5. (2 pts) Explain what is wrong with the following calculation of the area under the curve x4 from x =
0
5-4h44figb
1
2
1
2
Z 1
h is largest. Which is smallest? How many of the num1
1 i1
1 1
2
Figure 5.72
Figure 5.73
bers 3 positive?
are
dx =
=
=
.
x4
3x
3 3
3
1
1 45. (a) Without computing any integrals, explain why the
ins5-4h54-56
average value of f (x) the area [0, ] the curve
6. (3 pts) Suppose = sin x on undermust be between= 0 and x = b. Solve for a in terms of b.
x 0.5 to 1.
(b) Compute this average.
46. Figure ?? shows the standard normal distribution from
statistics, which is given by
2
1
e x /2 .
2
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. R For Problems ??, assuming F = f , mark the quantity on a 1 to x = 1: 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 x a b Figure 5.75 Z 5 1
dx = ln(5). Now ﬁnd a fraction which approximates ln(5), by
x
1
Z 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)
Z 3
1. Estimate
f (x) dx using R5 and L5 . −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, h0 (1) = 3, h00 (1) = 4, h(2) = 6, h0 (2) = 5, h00 (2) = 13, and h00 is
R2
continuous everywhere. Find 1 h00 (u) du. ...

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