Rewrite the expression without using the absolute value symbol. |1 - 2x2|
> 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 a method similar to that of Example 5, Solution 1 [start with (1 + i)4 – i4].
> 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
> 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
> The points of intersection of the cardioid r = 1 + sin θ and the spiral loop r = 2θ, -π/2 < θ < π/2, can’t be found exactly. Use a graphing device to find the approximate values of θ at which they intersect. Then use these values to estimate the area th
> Graph the function by starting with the graphs in Figures 13 and 14 and applying the transformations of Section 1.3 where appropriate. Figures 13: y = |sin x| 1 37 -1 (a) f(x) = sin x (b) g(x) 3 сos х + 1. kle +
> Convert from radians to degrees. (a). -7π/2 (b). 8π/4
> Graph the function by starting with the graphs in Figures 13 and 14 and applying the transformations of Section 1.3 where appropriate. Figures 13: 1 37 -1 (a) f(x) = sin x (b) g(x) 3 сos х + 1. kle + 1 y = 3 y=z tan(x 2
> Graph the function by starting with the graphs in Figures 13 and 14 and applying the transformations of Section 1.3 where appropriate. Figures 13: y = tan 2x 1 37 -1 (a) f(x) = sin x (b) g(x) 3 сos х + 1. kle +
> Graph the function by starting with the graphs in Figures 13 and 14 and applying the transformations of Section 1.3 where appropriate. Figures 13: 1 37 -1 (a) f(x) = sin x (b) g(x) 3 сos х + 1. kle + TT y = cos x 3
> Find all values of x in the interval [0, 2π] that satisfy the inequality. sin x > cos x
> Find all values of x in the interval [0, 2π] that satisfy the inequality. -1 < tan x < 1
> Find all values of x in the interval [0, 2π] that satisfy the inequality. 2 cos x + 1 > 0
> Find all values of x in the interval [0, 2π] that satisfy the inequality. sin x < 1/2
> Find all values of x in the interval [0, 2Ï€] that satisfy the equation. |tan x| = 1
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. -3 <<1
> Find all values of x in the interval [0, 2Ï€] that satisfy the equation. sin 2x = cos x
> Find all values of x in the interval [0, 2Ï€] that satisfy the equation. 2 sin'x = 1
> Convert from radians to degrees. (a). 4π (b). -3π/4
> Find all values of x in the interval [0, 2Ï€] that satisfy the equation. 2 cos x - 1 = 0
> If sin x = 1/3 and sec y = 5/4, where x and y lie between 0 and π/2, evaluate the expression. cos 2y
> If sin x = 1/3 and sec y = 5/4, where x and y lie between 0 and π/2, evaluate the expression. sin (x + y)
> Prove the identity. cos 3θ = 4 cos3θ - 3 cos θ
> Prove the identity. tan 2θ = 2 tan θ/ 1 – tan2θ
> Use Definition 4 to prove that limn→∞ n3 = ∞.
> Use Definition 3 to prove that if lim n→∞ |an| = 0, then limn→∞ an = 0.
> If a ball is thrown upward from the top of a building 128 ft high with an initial velocity of 16 ft/s, then the height above the ground seconds later will be h = 128 + 16t – 16t2 During what time interval will the ball be at least 32 ft above the ground?
> Use Definition 3 to prove that limn→∞ r n = 0 when |r| < 1.
> Prove the identity. sin (π/2 + x) = cos x
> Prove each equation. (a). Equation 14a (b). Equation 14b
> Convert from degrees to radians. (a). 3150 (b). 360
> (a). Determine how large we have to take x so that 1/x < 0.0001 (b). Use Definition 2 to prove that limx→∞ 1/x2 = 0.
> Find, correct to five decimal places, the length of the side labeled x. 22 сm х 8
> Find, correct to five decimal places, the length of the side labeled x. 8 cm
> Prove the statement using the ∈, δ definition of a limit and illustrate with a diagram like Figure 7. Figure 7: y A /y= 4x – 5 7+e 7 1-E 3-6 3+8 lim (tx + 3) = 2 エ→-2
> Prove the statement using the ∈, δ definition of a limit and illustrate with a diagram like Figure 7. Figure 7: y A /y= 4x – 5 7+e 7 1-E 3-6 3+8 lim (1 – 4x) = 13 I-3
> Given that limx→2 (5x – 7) = 3, illustrate Definition 1 by finding values of δ that correspond to ∈ = 0.1, ∈ = 0.05, and ∈ = 0.01.
> As dry air moves upward, it expands and in so doing cools at a rate of about C for each 100-m rise, up to about 12 km. (a). If the ground temperature is 200C, write a formula for the temperature at height h. (b). What range of temperature can be expected
> Use the given graph of f to find a number such that if 0 < |x - 5|< 8 \f(x) – 3|< 0.6 then y. 3.6 3 2.4 5 5 5.7 4.
> (a). Find a number δ such that if |x – 2| < δ, then |4x – 8| < ∈, where ∈ = 0.1. (b). Repeat part (a) with ∈ = 0.01.
> Find the exact trigonometric ratios for the angle whose radian measure is given. 4π/3
> Find the exact trigonometric ratios for the angle whose radian measure is given. 3π/4
> Draw, in standard position, the angle whose measure is given. (a) rad 3 (b) —3 гad
> Use the given graph of f (x) = 1/x to find a number δ such that if |x - 2| < 8 then 0.5 < 0.2 y. 1+ 0.7 0.5 0.3 10 10 3 2.
> Draw, in standard position, the angle whose measure is given. 37 rad 4 (а) 315° (b)
> If a circle has radius 10 cm, find the length of the arc subtended by a central angle of 720.
> Find the length of a circular arc subtended by an angle of π/12 rad if the radius of the circle is 36 cm.
