Prove formula (e) of Theorem 3 using a method similar to that of Example 5, Solution 1 [start with (1 + i)4 – i4].
> One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. If represents the number of female insects in a popul
> (a). Show that the area of a triangle with sides of lengths a and b and with included angle θ is A = 1/2 ab sin θ (b). Find the area of triangle ABC, correct to five decimal places, if |AB| = 10 cm | BC| = 3 cm ZABC 107°
> Graph both y = 1/ (x3 – 2x2) and an antiderivative on the same screen.
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 2x + 1 (a) x? + 4x + 3 (b) (x + 1)°(x² + 4)²
> Use a graph of f (x) = 1/ (x2 – 2x – 3) to decide whether f20 f (x) dx is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.
> Make a substitution to express the integrand as a rational function and then evaluate the integral. cos x dx sin'x + sin x sin*x
> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx e2* + 3e* + 2
> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx 2/x + 3 + x
> Make a substitution to express the integrand as a rational function and then evaluate the integral. x, dx J9 x- 4 16
> Evaluate the integral. f 3x2 + x + 4/ x4 + 3x2 + 2, dx
> Evaluate the integral. f x – 3/ (x2 + 2x + 4)2, dx
> Evaluate the integral. f x4 + 3x2 + 1/x5 + 5x3 + 5x, dx
> (a). If is a complex-valued function of a real variable, its indefinite integral f u (x) dx is an antiderivative of u. Evaluate (b). By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals (c). Compare wit
> Evaluate the integral. f dx/x (x2 + 4)2
> Evaluate the integral. f x3/x3 + 1, dx
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. x* + 1 (a) x* + 4x 1 (b) (x² – 9)²
> Evaluate the integral. f 1/x3 – 1, dx
> Evaluate the integral. f10 x/x2 + 4x + 13, dx
> Evaluate the integral. f x + 4/x2 + 2x + 5, dx
> Evaluate the integral. f x2 + x + 1/ (x2 + 1)2, dx
> Evaluate the integral. f x3 + x2 + 2x + 1/ (x2 + 1) (x2 + 2), dx
> Evaluate the integral. f x2 – 2x – 1/ (x – 1)2 (x2 + 1), dx
> Evaluate the integral. f 10/ (x – 1) (x2 + 9), dx
> Suppose an ellipse has foci (±c, 0) and the sum of the distances from any point P (x, y) on the ellipse to the foci is 2a. Show that the coordinates of P satisfy Equation 1.
> Use the given graph of f (x) = √x to find a number δ such that if |x – 4|< 8 |V - 2|<0.4 then y4 y= Va 2.4. 2 1.6 4
> Evaluate the integral. f x2 – x + 6/x3 + 3x, dx
> Evaluate the integral. f 5x2 + 3x – 2/x3 + 2x2, dx
> Evaluate the integral. f x2 – 5x + 16/(2x + 1) (x – 2)2 dx
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. x? (a) x? + x – 2 (b) x? + x + 2
> Evaluate the integral. f 1/(x + 5)2 (x – 1), dx
> Evaluate the integral. f x2 + 2x – 1/x3 – x, dx
> Evaluate the integral. f11 4y2 – 7y -12/y (y + 2) (y – 3), dy
> Evaluate the integral. f10 x3 – 4x – 10/x2 – x – 6, dx
> Evaluate the integral. f43 x3 – 2x2 - 4/x3 – 2x2, dx
> Evaluate the integral. f 1/(x + a) (x + b), dx
> Match the polar equations with the graphs labeled I–VI. Give reasons for your choices. (Don’t use a graphing device.) I II III IV V VI (a) r = Ve, 0<0 < 167 (b) r= 0², 0< 0< 167 (c) r= cos(0/3) (e) r = 2 + sin 30
> Evaluate the integral. f10 x - 1/x2 + 3x + 2, dx
> Evaluate the integral. f32 1/x2 – 1, dx
> Evaluate the integral. f 1/(t – 4) (t – 1), dt
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 2x 1 (a) (x + 3)(3x + 1) (b) x + 2x? + x
> Write the sum in expanded form. ∑n+3j=n j2
> Write the sum in expanded form. ∑ni=1 i10
> Write the sum in expanded form. ∑8k=5 xk
> Write the sum in expanded form. ∑4k=1 2k – 1/ 2k + 1
> Prove the generalized triangle inequality: |∑ n i=1 ai| < ∑n i=1 |ai|
> Evaluate each telescoping sum. (a). ∑n i=1 [i4 – (i – 1)4] 100 (b) E (5' – 5i-1) i- 99 (c) Σ (d) Σ (a, - a-1) i+1 i-3 i-l
> (a). In Example 11 the graphs suggest that the limaçon r = 1 + c sin θ has an inner loop when |c| > 1. Prove that this is true, and find the values of θ that correspond to the inner loop. (b). From Figure 19 it a
> Prove formula (e) of Theorem 3 using the following method published by Abu Bekr Mohammed ibn Alhusain Alkarchi 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, . . . n, Thus the s
> Write the sum in expanded form. ∑6i=4 i3
> Prove formula (e) of Theorem 3 using mathematical induction.
> Prove formula (b) of Theorem 3.
> Find the number n such that ∑n i=1 i = 78.
> Find the value of the sum. ∑ni=1 (t3 – i - 2)
> Find the value of the sum. ∑ni=1 (i + 1) (i + 2)
> Find the value of the sum. ∑ni=1 (i + 1) (i + 2)
> Find the value of the sum. ∑ni=1 (3 + 2i)2
> If f is a quadratic function such that f (0) = 1 and f f (x)/ x2 (x + 1)3, dx is a rational function, find the value of f'(0).
> Find the value of the sum. ∑ni=1 (i2 + 3i + 4)
> Find the value of the sum. ∑ni=1 (2 – 5i)
> Write the sum in expanded form. ∑6i=4 3i
> Find the value of the sum. ∑ni=1 2i
> Find the value of the sum. ∑4i=-2 23-i
> Find the value of the sum. ∑4i=0 (2i + i2)
> Find the value of the sum. ∑4i=1 4
> Find the value of the sum. ∑20n=1 (-1)n
> Find the value of the sum. ∑8k=0 cos kπ
> Find the value of the sum. ∑6j=4 3i + 1
> Suppose that F, G and Q are polynomials and F (x)/ Q (x) = G (x)/ Q (x) for all x except when Q (x) = 0. Prove that F (x) = G (x) for all x. [Hint: Use continuity.]
> Find the value of the sum. ∑6i=3 i(i + 2)
> Find the value of the sum. ∑8i=4 (3i – 2)
> Write the sum in sigma notation. 1 – x + x2 – x3 + … + (-1)n xn
> Write the sum in expanded form. ∑6i=1 1/i +1
> Write the sum in sigma notation. x + x2 + x3 + … + xn
> Write the sum in sigma notation. 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36
> Write the sum in sigma notation. 1 + 2 + 4 + 8 + 16 + 32
> Write the sum in sigma notation. 1 + 3 + 5 + 7+… + (2n – 1)
> Write the sum in sigma notation. 2 + 4 + 6 + 8 +… + 2n
> Write the sum in sigma notation. 3/7 + 4/8 + 5/9 + 6/10 + … 23/27
> (a). Find the partial fraction decomposition of the function (b). Use part (a) to find ff (x) dx and graph f and its indefinite integral on the same screen. (c). Use the graph of f to discover the main features of the graph of ff (x) dx. 12x – 7x'
> Write the sum in sigma notation. 1/2 + 2/5 + 3/4 + 4/5 + … + 19/20
> Write the sum in sigma notation. √3 + √4 + √5 + √6 + √7
> Write the sum in sigma notation. 1+ 2 + 3 + 4 + … + 10
> Write the sum in expanded form. ∑nj=1 f (xi) ∆xi
> Write the sum in expanded form. ∑5i=1 √i
> Use Definition 1 to prove that limx→0 x3 = 0
> Find the radius of a circular sector with angle 3π/4 and arc length 6 cm.
> A circle has radius 1.5 m. What angle is subtended at the center of the circle by an arc 1 m long?
> Use a graph to find a number δ such that if 2x |x - 1|< 8 then - 0.4 <0.1 x? + 4
> Find the foci of the ellipse x2 + 4y2 = 4 and sketch its graph.
> (a). Use a computer algebra system to find the partial fraction decomposition of the function (b). Use part (a) to find ff (x) dx (by hand) and compare with the result of using the CAS to integrate f directly. Comment on any discrepancy. 4x3 — 27х?
> Use a graph to find a number δ such that if < 8 4 | tan x – 1| < 0.2 then
> Rewrite the expression without using the absolute value symbol. |1 - 2x2|
> Find the focus and directrix of the parabola y = x2. Illustrate with a diagram.
> Rewrite the expression without using the absolute value symbol. |5 – 23|
> (a). Show that if the - and -intercepts of a line are nonzero numbers a and b, then the equation of the line can be put in the form x/a + y/b = 1 This equation is called the two-intercept form of an equation of a line. (b). Use part (a) to find an equati
> Use the addition formula for cosine and the identities to prove the subtraction formula (13a) for the sine function. (÷-)- : )-i : )- sin 2 cos sin e cos e
> Use the formula in Exercise 43 to prove the addition formula for cosine (12b). Exercise 43: Use the figure to prove the subtraction formula cos (α – β) = cos α cos β + sin Î&pl
> Use the figure to prove the subtraction formula cos (α – β) = cos α cos β + sin α sin β [Hint: Compute c2 in two ways (using the Law of Cosines from Exercise
> In order to find the distance |AB| across a small inlet, a point C was located as in the figure and the following measurements were recorded: Use the Law of Cosines from Exercise 41 to find the required distance. LC = 103° |AC|= 820 m | BC| = 910
> Prove the Law of Cosines: If a triangle has sides with lengths a, b, and c, and θ is the angle between the sides with lengths a and b, then [Hint: Introduce a coordinate system so that θ is in standard position, as in the fig
