Homework 6-Integration

¨ University Faculty of Science Department of Mathematics Dokuz Eylul

MAT 1031 Calculus I

18.12.2013

Homework 6 - Integration k: [email protected]

˘ Instructors: Celal Cem Sarıoglu

k: [email protected]

Didem Cos¸kan

Course Web-page:

Z: B351/2

T: 0 232 3018607

Z: B351/1

T: 0 232 3018608

http://kisi.deu.edu.tr/celalcem.sarioglu/13gmat1031.html

Textbook: University Calculus, Early Transcendentals, Joel Hass, Maurice D. Weir, and George B. Thomas, Jr., International Edition, 2nd edition, Pearson, 2012.

Reading: Read the following parts from the Calculus Biographies that I have given (online supplement of our textbook): 1. History of the Integral 2. The Fundamental Theorem of Calculus 3. History of Differential Equations 4. Biographies of the following mathematicians (and scientists): (a) Carl Friedrich Gauss (1777-1855)

(c) George David Birkhoff (1884-1944)

(b) Georg Friedrich Bernhard Riemann (1826-1866)

(d) Richard Dedekind (1831-1916)

5. After you have studied our textbook, you may in addition use the following issues of Matematik Dnyası which contain some subjects of integration that we shall study this term: (a) MD 2011-II, pages 27–67.

(d) MD 2012-I, pages 21–23.

(b) MD 2011-III, pages 44–77.

(e) MD 2012-II, pages 33–36.

(c) MD 2011-IV, pages 24–30, 39–50.

(f) MD 2012-III, pages 27–61.

Solve the Below Exercises. 1. Use finite approximations to estimate the area under the graph of the function f(x) = 4 − x2 between x = −2 and x = 2, using (a) a lower sum with four rectangles of equal width (b) an upper sum with four rectangles of equal width

(a)

(d)

k

k=1

(b)

100 P

(c)

6. Let f(x) be a continuous function. Express 1 1 2 3 n lim f( ) + f( ) + f( ) + · · · + f( ) n→∞ n n n n n

7. Evaluate 15 + 25 + 35 + · · · + n5 n→∞ n6

k(3k + 5)

lim

k=1

k2

(e)

k=1 100 P

100 P

n→∞

compare the area of disk of radius r.

as a definite integral.

2. Evaluate the following finite sums 100 P

gruent triangles. Compute the area An of the inscribed polygon. Compute the limit lim An , and

n P

nk(k2 + 3k − n)

k=1

k

n P 1 (f) ( + 2n) n k=1

3

k=1

3. Find the norm of the partition P = {−2, −1.6, −0.5, 0, 0.8, 1}. Pn 2 4. Express the limit lim k=1 (ck − 3ck )∆xk , where ||P||→0

P is a partition of [−7, 5]. 5. Inscribe a regular n-sided polygon inside a circle of radius r and compute the area of one of the n con-

by showing that the limit is

R1

x5 dx.

0

8. Evaluate the following limits: 13 + 23 + 33 + · · · + n3 . n→∞ n4 2 + 4 + 6 + · · · + 2n (b) lim . n→∞ n2 1 π 2π 3π nπ (c) lim sin + sin + sin + · · · + sin . n→∞ n n n n n 1 1 2 3 n−1 2 (d) lim ( )2 + ( )2 + ( )2 + · · · ( ) . n→∞ n n n n n (a) lim

MAT 1031 Calculus I – Homework 6 - Integration,

˘ & Didem Cos¸kan Instructors: Celal Cem Sarıoglu

Date: 18.12.2013

9. Graph the integrands and use areas to evaluate the following integrals: Z2 Z1 p c) (1 + 4 − x2 ) dx a) (1 − |x| ) dx −1 Z0

b)

p

−2 Z2

4 − x2 dx

d)

(x +

−2

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To make consistent these two concepts, it will be reasonable to have f(b) − f(a) = av(f 0 ), b−a

p 4 − x2 ) dx

that is to have, the average rate of change of f on [a, b] equals the average value of f 0 on [a, b]. Is this the case? Give reasons for your answer.

−2

10. What values of a and b maximize the value of Zb (x − x2 ) dx ? (Hint: Where is the integrand posia

18. Another motivation for the definition of the average value of a continuous function y = f(x) on an interval [a, b] is as follows. Let n ∈ Z+ . Make a partition of [a, b] by dividing it into n intervals of length (b − a)/n, that is, take the partition

tive?) 11. What values of a and b minimize the value of Zb 4 (x − 2x2 ) dx ? a

12. Use the Max-Min inequality to find upper and Z1 1 lower bounds for the value of dx. 1 + x2 0

P = {x0 , x1 , x2 , . . . , xn−1 , xn }, where xk = a + k

13. Use the Max-Min inequality to show that if f is integrable then Zb (a) f(x) > 0 on [a, b] =⇒ f(x) dx > 0

b−a for k = 0, 1, 2, . . . , n. Then n

∆xk = xk − xk−1 =

b−a n

a

Zb (b) f(x) 6 0 on [a, b]

for all k = 1, 2, . . . , n, and so the norm of the partition P is

f(x) dx 6 0

=⇒ a

14. Use the inequality sin x 6 x, which holds for x > 0, R1 to find an upper bound for the value of sin x dx.

kPk = max({∆x1 , ∆x2 , . . . , ∆xn }) =

b−a . n

0

Choose a point ck from each interval [xk−1 , xk ] for k = 0, 1, 2, . . . , n. The average value of the n values f(c1 ), f(c2 ), . . . , f(cn ) is

15. If av(f) really is a typical value of the integrable function f(x) on [a, b], then the constant function av(f) should have the same integral over [a, b] as f. Does it? That is, does Zb Zb av(f) dx = f(x) dx ? a

f(c1 ) + f(c2 ) + · · · + f(cn ) . n

a

What is the limit of this as n → ∞ ? (Hint: Is this like a Riemann sum?)

Give reasons for your answer. 16. It would be nice if average values of integrable functions obeyed the following rules on an interval [a, b] :

19. Is it true that every function y = f(x) that is differentiable on [a, b] is itself the derivative of some function on [a, b]? Give reasons for your answer.

(a) av(f + g) = av(f) + av(g) (b) av(kf) = k av(f), where k is a real constant.

20. Use the formula

(c) av(f) 6 av(g) if f(x) 6 g(x) on [a, b]

m X

Do these rules ever hold? Give reasons for your answer.

k=1

17. Let f be a function that is differentiable on [a, b]. We have defined the average rate of change of f on [a, b] to be f(b) − f(a) , b−a and the instantaneous rate of change of f at each x ∈ [a, b] to be f 0 (x). On the other hand, for a function g that is continuous on [a, b], we have defined the average value av(g) of g on the interval [a, b] to be Zb 1 av(g) = g(x) dx. b−a

cos h2 − cos (m + 12 )x sin kx = 2 sin x2

to find the area under the curve y = sin x from x = 0 to x = π/2 in two steps: (a) Partition the interval [0, π/2] into n sub-intervals of equal length and calculate the corresponding upper sum U; then (b) Find the limit of U as n → ∞ and b−a ∆x = → 0. n 21. Suppose F(x) is an antiderivative of f(x) = Z3 sin 2x x > 0. Express dx in terms of F. x 1

a

2

sin x , x



Date: 18.12.2013

Zx

22. Evaluate the following integrals: Z4 Z1 √ 2 (a) (x + x) dx (h) xπ−1 dx 0 2 Z 32 Z0 6 (b) x− 5 dx (i) πx−1 dx 1 Z π/4 −1 Z5 (c) tan2 x dx x 0 √ dx (j) Z −1 5 1 + x2 y − 2y 2 (d) dy Z π/3 y3 −3 Z4 (k) sin2 x cos x dx 0 (e) |x| dx −4 Z π/6 Z ln 2 (l) (sec x + tan x)2 dx (f) e3x dx 0 Z 1/√3

(g) 0

(h) y = −1

Zπ (m) 0

f(t) dt = x2 − 2x + 1. 1

Zx 27. Find f(4) if

f(t) dt = x cos(πx). 0

28. Suppose that f has a positive derivative for all values of x and that f(1) = 0. Which of the following Zstatements must be true of the function

cos x + | cos x| dx 2

x

f(t) dt ? Give reasons for your answer.

g(x) = 0

(a) g is a differentiable function of x. (b) g is a continuous function of x. (c) g has a local minimum at x = 1. (d) g has a local maximum at x = 1.

dy 24. Find the derivative of the following functions: dx Zx Z sin x sint dt (a) y = dt (e) y = t 1 − t2 cos x Z1tan x sec2 t dt (b) y = Z √

(e) The graph of g has a horizontal tangent at x = 1. (f) The graph of g has an inflection point at x = 1. (g) The graph of dg/dx crosses the x-axis at x = 1.

2 x

0 Z x3

(f) y = e−t dt

0

Z ex2 0

t2 dt +4

Zx 26. Find f(x) if

u(x)

(d) y =

3

t2

x

23. Prove the Leibniz’s Rule: if f is continuous on [a, b] and if u(x) and v(x) are both differentiable functions of x whose values lie in [a, b], then   v(x) Z du d  dv  − f(u(x)) . f(t) dt = f(v(x))  dx dx dx

(c) y =

Zx

25. Use Leibniz’s Rule to find the value Z x+3 of x that maximizes the value of the integral t(5 − t) dt.

0

dx 1 + 4x2

t2 dt − 2 t +4

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1 √ dt t

2

√ x

sin(t ) dt 29. Using the derivative of hyperbolic functions and inverse hyperbolic functions, prove the integration formulas given in Table 7.7 and and Table 7.10 in your textbook (pages 421 and 425).

Z 2x4 (g) y =

sin x cos t dt x2

Solve the Below Exercises from your Textbook. Of course, solve as many exercises as you need to be sure that you have learned the concepts and can do computations without error but I require you to be prepared to solve some of the following exercises next week: Sec. 4.8 Antiderivatives (pages 277-281) Exercises: 2, 4, 7, 8, 14–19, 21–24, 39–70, 72, 73, 76–91, 98, 104–111, 113–118, 121, 125, 127, 128. Sec. 5.1 Area and Estimating with Finite Sums (pages 296-298) Exercises: 1–11, 14, 21, 22. Sec. 5.2 Sigma Notation and Limits of Finite Sums (pages 304-305) Exercises: 1–32, 38, 43. Sec. 5.3 The Definite Integral (pages 313-317) Exercises: 1–9, 11–13, 15–22, 27, 28, 41–50, 55, 56, 61, 62, 71–86, 89–93. Sec. 5.4 The Fundamental Theorem of Calculus (pages 325-328) Exercises: 7–18, 21–64, 69–73, 75–84 Sec. 7.2 Hyperbolic Functions (pages 425-428) Exercises: 1–12, 14, 15, 17, 20, 23, 25–27, 29, 31–82, 85, 86. We have already seen hyperbolic functions and inverse hyperbolic functions, their graphs and derivatives. Just make a review of them. Sec. 7.1 The Logarithm Defined as an Integral (pages 409-411) Exercises: 1–9, 12–14, 23–27, 33–36, 40, 41, 47, 48, 51, 53–56, 59, 60, 64, 67, 70

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Date: 18.12.2013

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Study Section 7.1 independently to learn how one can define rigorously the transcendental functions ln x and ex , and obtain all of their usual properties. A few weeks later you have solved these exercises, to keep fresh your knowledge it is suggested to solve the following exercises from your textbook on pages 345–352 and 428–430: • Questions to Guide your Review • Practice Exercises • Additional and Advanced Exercises

Matematik Dunyası: ¨ http://www.matematikdunyasi.org/ ¨ It is suggested to subscribe this magazine. The following issues of the Matematik Dunyası magazine are related with our courses MAT 1031 Calculus I and MAT 1032 Calculus II: ˘ Sayılar) • MD 2003-IV 2 × 2 = 4 (Dogal

• MD 2010-III Fonksiyon çizimi,

• MD 2006-IV: Tamsayılar, Kesirli Sayılar ve Sıralı Halkalar

˘ çizimleri, • 2010-IV Fonksiyon ve Egri

• MD 2007-I: Kesirli Sayı Dizileri ve Gerçel Sayılar (I) ˙ ¸ ası (II) • MD 2007-II: Gerçel Sayıların Ins

˙ • MD 2011-II Integral I, ˙ • MD 2011-III Integral II,

• MD 2007-III Analiz I: Diziler,

˙ • MD 2011-IV Integral III,

• MD 2007-IV Analiz II: Diziler ve exp fonksiyonu,

• MD 2012-I Analizden Konular,

• MD 2008-I Analiz III: Seriler,

• MD 2012-II Analizden Konular II,

• MD 2008-II Analiz IV: Seriler, ¨ • MD 2008-III Analiz V: Sureklilik ve Limit,

˙ • MD 2012-III Integral IV,

¨ un ¨ Yakınsaklık, • MD 2008-IV Analiz-V: Duzg

• MD 2012-IV Analizden Konular III,

¨ • MD 2010-II Turev,

• MD 2013-I Analizden Konular IV.

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