Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Single Variable Calculus: Concepts and Contexts, Enhanced Edition
1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E1EWrite the sum in expanded form. 2. i=161i+13EWrite the sum in expanded form. 4. i=46i35E6E7E8EWrite the sum in expanded form. 9. j=0n1(1)jWrite the sum in expanded form. 10. i=1nf(xi)xi11EWrite the sum in sigma notation. 12. 3+4+5+6+713EWrite the sum in sigma notation. 14. 37+48+59+610++232715E16E17E18E19E20E21E22E23E24E25EFind the value of the sum. 26. i=1100427E28E29E30EFind the value of the sum. 31. i=1n(i2+3i+4)32E33E34E35E36E37E38E39EProve formula (e) of Theorem 3 using the following method published by Abu Bekr Mohammed ibn Alhusain Alkar-chi in about ad 1010. The figure shows a square ABCD in which sides AB and AD have been divided into segments of lengths 1, 2, 3, , n. Thus the side of the square has length n(n + 1)/2 so the area is [n(n + 1)/2]2. But the area is also the sum of the areas of the n gnomons G1, G2, ..., Gn shown in the figure. Show that the area of Gi is i3 and conclude that formula (e) is true.Evaluate each telescoping sum. (a) i=1n[i4(i1)4] (b) i=1100(5i5i1) (c) i=399(1i1i+1) (d) i=1n(aiai1)42E43E44E45E46E47E48E49E50E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E69E70E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E1E2E3E4E5E6E7EEvaluate the expression and write your answer in the form a + bi. 8. 3+2i14i9E10E11E12E13EEvaluate the expression and write your answer in the form a + bi. 14. 31215E16E17EProve the following properties of complex numbers. (a) z+w=z+w (b) zw=zw (c) zn=zn, where n is a positive integer [Hint: Write z = a + bi, w = c + di.]19E20E21EFind all solutions of the equation. 22. 2x2 2x + 1 = 023EFind all solutions of the equation. 24. z2+12z+14=0Write the number in polar form with argument between 0 and 2. 25. 3 + 3i26E27E28EWrite the number in polar form with argument between 0 and 2. 29. z=3+i,w=1+3i30E31E32EFind the indicated power using De Moivres Theorem. 33. (1 + i)2034E35E36E37E38E39E40E41E42E43E44E45E46E47EUse Eulers formula to prove the following formulas for cos x and sin x: cosx=eix+eix2sinx=eixeix2iIf u(x) = f(x) + ig(x) is a complex-valued function of a real variable x and the real and imaginary parts f(x) and g(x) are differentiable functions of x, then the derivative of u is defined to be u(x) = f(x) + ig(x). Use this together with Equation 7 to prove that if F(x) = erx, then F(x) = rerx when r = a + bi is a complex number.(a) If u is a complex-valued function of a real variable, its indefinite integral u(x)dx is an antiderivative of u. Evaluate e(1+i)xdx (b) By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals excosxdxandexsinxdx (c) Compare with the method used in Example 7.1.4.