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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). 1 lim x-4 √x-2 x 4 - 4/
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). 1 lim x-11-x X 2 1-x²
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). 1 lim X→0+ √x 1 1 √x² + x
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). cot x lim x-0 CSC X
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim 6- cot 0 csc 0
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim 1-2 22 +2¹-20 2¹ - 4
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim X- 2 cos²x + 3 cos x - 2 2 cos x - 1
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim klt 1 tan 0-1 2 tan2²0 - 1
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim (sec 0 tan 0) 8→
Use a plot of to estimate to two decimal places. Compare with the answer obtained algebraically in Exercise 23.Data From Exercise 23Evaluate the limit, if it exists. If not, determine whether the
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim x-4 KI+ sin x - cos x tan x-1
The following limits all have the indeterminate form 0/0. One of the limits does not exist, one is equal to 0, and one is a nonzero limit. Evaluate each limit algebraically if you can or investigate
Show that the limit is in an indeterminate form, then investigate the limit numerically to estimate the value. lim 1-cos² 0² 0→0
The following limits all have the indeterminate form ∞/∞. One of the limits does not exist, one is equal to 0, and one is a nonzero limit. Evaluate each limit algebraically if you can or
Show that the limit is in an indeterminate form, then investigate the limit numerically to estimate the value. lim 1-cos e 0-0 82
Use a plot of to estimate numerically. Compare with the answer obtained algebraically in Exercise 25.Data From Exercise 25Evaluate the limit, if it exists. If not, determine whether the one-sided
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate in terms of the constant a. lim (4ah + 7a) h→→2
Evaluate using the identity a³-b³ = (a - b)(a² + ab + b²)
Evaluate in terms of the constant a. lim(2a + x) X-0
Evaluate in terms of the constant a. lim (4t-2at + 3a) 14-1
Evaluate in terms of the constant a. lim x→a (x + a)² - 4x² x - a
Let ƒ(x) = (x − 100)2 + 1000. What will the display show if you graph ƒ in the viewing rectangle [−10, 10] by [−10, 10]? Find an appropriate viewing rectangle.
Describe the pairs of numbers x, y such that |x + y| = x − y.
Plot ƒ(x) = 8x + 1/8x − 4 in an appropriate viewing rectangle. What are the vertical and horizontal asymptotes?
Fill in the remaining values of (cos θ, sin θ) for the points in Figure 21. 5% К Но 6 3П 4 SA 4 2п 3 4% 3 kle 3П 2 4 7 5П 4 3 6 2 0 (1,0) 11 6
How many solutions does x3 − 4x + 8 = 0 have?
According to the evidence in Example 4, it appears that ƒ(n) = (1 + 1)n never takes on a value greater than 3 for n > 0. Does this evidence prove that ƒ(n) ≤ 3 for n > 0?Example No 4 OF
Indicate which of the following are correct, and correct the ones that are not. (a) 52.51/2 = 5 (c) 38 34 = = 3² (b) (√8)4/3 = 82/3 (d) (24) −2 = 22
Describe θ = π/6 by an angle of negative radian measure.
Convert from radians to degrees: (a) 1 (b) (c) 5/2 (d) - 3
How many positive solutions does x3 − 12x + 8 = 0 have?
Express (4, 10) as a set {x : |x − a| < c} for suitable a and c.
How can a graphing calculator be used to find the minimum value of a function?
Convert from degrees to radians: (a) 1° (b) 30° (c) 25° (d) 120°
Does cos x + x = 0 have a solution? A positive solution?
Express as an interval:(a) {x : |x − 5| < 4} (b) {x : |5x + 3| ≤ 2}
Find the lengths of the arcs subtended by the angles θ and ϕ radians in Figure 19. 4 = 2 0=0.9
Find all the solutions of sin x = √x for x > 0.
Express {x : 2 ≤ |x − 1| ≤ 6} as a union of two intervals.
Give an example of numbers x, y such that |x| + |y| = x − y.
Find the values of the six standard trigonometric functions at θ = 11π/6.
(a) Plot the graph of ƒ(x) = 2x2 + x/1−x2 in a viewing rectangle that clearly shows the two vertical asymptotes.(b) By examining the graph of ƒ in appropriate viewing rectangles, approximate
Assume that 0 ≤ θ < π/2.Find sin θ, cos θ, and secθ if cot θ = 4.
Find tan θ if sec θ = √5 and sinθ < 0.
Determine a function that would have a graph as in Figure 25(A), stating the period and amplitude. 3 N -π 2п 3π -3 (A) X
Sketch the graph by hand.y = x1/3
Determine a function that would have a graph as in Figure 25(B), stating the period and amplitude. E ان انيا (B) اس X
Sketch the graph by hand.y = 1/x2
Show that the graph of y = ƒ(1/3 x − b) is obtained by shifting the graph of y = ƒ(1/3 x) to the right 3b units. Use this observation to sketch the graph of y = Ι1/3 x − 4Ι.
Let h(x) = cos x and g(x) = x−1. Compute the composite functions h ◦ g and g ◦ h, and find their domains.
During a year, the length of a day, from sunrise to sunset, in Taloga, Oklahoma, varies from a shortest day of approximately 9.6 hours to a longest day of approximately 14.4 hours, while in Montreal,
Sketch the points on the unit circle corresponding to the following three angles, and find the values of the six standard trigonometric functions at each angle: (a) 3 (b) 4 (c) 7x 6
How many points lie on the intersection of the horizontal line y = c and the graph of y = sin x for 0 ≤ x < 2π? The answer depends on c.
How many points lie on the intersection of the horizontal line y = c and the graph of y = tan x for 0 ≤ x < 2π?
What are the periods of these functions?(a) y = sin 2θ (b) y = sin θ/2(c) y = sin 2θ + sin θ/2
Determine A, B, and C so that ƒ(x) = A cos(Bx) + C cycles once from 8 to −2 and back to 8 as x goes from 0 to 2.
Derive the identity using the identities listed in this section.cos 2θ = 2 cos2 θ − 1
H(t) = A sin(Bt) + C models the height (in meters) of the tide in Happy Harbor at time t (hours since midnight) in a day. Determine A, B, and C if the high tide of 18 m occurs at 6:00 A.M. and the
Derive the identity using the identities listed in this section.cos2 θ/2 = 1 + cos θ/2
Assume that sin θ = 4/5, where π/2 < θ < π. Find:(a) tan θ (b) sin 2θ (c) csc θ/2
Derive the identity using the identities listed in this section.sin2 θ/2= 1 − cos θ/2
Give an example of values a, b such that (a) cos(a + b) # cos a + cos b (b) cos # cos a 2
Derive the identity using the identities listed in this section.sin(θ + π) = − sin θ
Let ƒ(x) = cos x. Sketch the graph of y = 2ƒ(1/3 x − π/4) for 0 ≤ x ≤ 6π.
Derive the identity using the identities listed in this section.cos(θ + π) = − cos θ
Solve sin 2x + cos x = 0 for 0 ≤ x < 2π.
Derive the identity using the identities listed in this section.tan x = cot(π/2 − x)
How does h(n) = n2/2n behave for large whole-number values of n? Does h(n) tend to infinity?
Use a graphing calculator to determine whether the equation cos x = 5x2 − 8x4 has any solutions.
Derive the identity using the identities listed in this section.tan 2x = 2 tan x/1 − tan2x
Derive the identity using the identities listed in this section. sin² x cos²x = 1- cos 4x 8
Using a graphing calculator, find the number of real roots and estimate the largest root to two decimal places: (a) f(x) = 1.8x4 - x² - x (b) g(x) = 1.7x4-x5-x
Sketch the graph of y = ƒ(x + 2) − 1, where ƒ(x) = x2 for −2 ≤ x ≤ 2.
Let ƒ (x) be the function shown in Figure 1.Sketch the graphs of y = ƒ(x) + 2 and y = ƒ(x + 2). 3 2 1 0 y 1 2 3 نیا 4 X
Plot the graph of ƒ(x) = x/(4 − x) in a viewing rectangle that clearly displays the vertical and horizontal asymptotes.
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition.cos θ = 1/2 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
Let ƒ (x) be the function shown in Figure 1.Sketch the graphs of y = 1/2 ƒ(x) and y = ƒ (1/2 x). 3 2 1 0 y 1 2 3 نیا 4 X
Illustrate local linearity for ƒ(x) = x2 by zooming in on the graph at x = 0.5 (see Example 6). Hypotenuse 5 FIGURE 16 FIGURE 17 a Adjacent 2 (o Opposite √21 EXAMPLE 6 For between 0 and 27, the
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition. tan θ = 1 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
Let ƒ (x) be the function shown in Figure 1.Continue the graph of ƒ to the interval [−4, 4] as an even function. 3 2 1 0 y 1 2 3 نیا 4 X
Plot ƒ(x) = cos(x2) sin x for 0 ≤ x ≤ 2π. Then illustrate local linearity at x = 3.8 by choosing appropriate viewing rectangles.
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition. tan θ = −1 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
Let ƒ (x) be the function shown in Figure 1.Continue the graph of ƒ to the interval [−4, 4] as an odd function. 3 2 1 0 y 1 2 3 نیا 4 X
By zooming in on the graph of ƒ(x) = √x at x = 0, examine the local linearity. How does the resulting “line” appear?
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition.csc θ = 2 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
Investigate the behavior of the function as n or x grows large by making a table of function values and plotting a graph (see Example 4). Describe the behavior in words.ƒ(n) = (1 + 1)n2 -1- OF
By examining the graph of ƒ(x) = 2x2 − x3 − 3x4 in appropriate viewing rectangles, approximate the maximum value of ƒ(x) and the value of x at which it occurs.
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition. sin x = √3/2 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
Use Figure 21 to find all angles between 0 and 2π satisfying the given condition. sec t = 2 Sn П 3п 4 5 4 2п 3 4 3 k/e Зл 2 3 2 П 4 7 п 3л 4 3 62' 4) -0 (1,0) 11z 6
If $500 is deposited in a bank account paying 3.5% interest compounded monthly, then the account has value V(N) = 500 (1 + 0.035/12) N dollars after N months. By examining the graph of V(N), find,
Complete the following table of signs: 0 < 0 < π 2 0 3π 2 T]
If $1000 is deposited in a bank account paying 5% interest compounded monthly, then the account has value V(N) = 1000 (1 + 0.05/12)N dollars after N months. By examining the graph of V(N), find, to
Determine whether the function is increasing, decreasing, or neither: (a) f(x) = √8-x (c) g(t) = t¹² + t (b) f(x) = 2+1 (d) g(t) = t³ + t
Investigate the behavior of the function as n or x grows large by making a table of function values and plotting a graph (see Example 4). Describe the behavior in words.ƒ(n) = n1 -1- OF FIGURE
Determine whether the function is even, odd, or neither:(a) ƒ(x) = x4 − 3x2(b) g(x) = sin(x + 1)

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