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mathematics
calculus

Questions and Answers of Calculus

Sketch the graphs of the following on [-π, 2π]. (a) y = sin 2x (b) y = 2 sin t (c) y = cos(x - π/4) (d) y = sec t
Sketch the graphs of the following on [-π, 2π]. (a) y = csc t (b) y = 2 cos t (c) y = cos 3t (d) y = cos (t + π/3)
Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval - 5 ≤ x ≤ 5 for the functions listed in Problems 1-3 1. y = 3 cos x/2 2. y = 2 sin
Convert the following radian measures degrees. (a) 7/6 π (b) 3/4 π (c) - 1/3 π (d) 4/3 π (e) -35/18 π (f) 3/18 π
Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) y = sin(x + π/2) (b) y = cos(x + π/2) (c) y = -sin(x + π) (d) y = cos (x -
Which of the following are odd function? Even function? Neither? (a) t sin t (b) sin2 t (c) csc t (d) |sin t| (e) sin (csc t) (f) x + sin x
Which of the following are odd function? Even functions Neither? (a) cot t + sin t (b) sin3 t (c) sec t (d) √sin4 t (e) cos (sin t) (f) x2 + sin x
Find the exact values in problem 1-5. 1. cos2 π/3 2. sin2 π/6 3. sin3 π/6 4. cos2 π/12 5. sin2 π/8
Convert the following degree measures to radians (1° = π/180 ≈ 1.7453 × 10-2 radian). (a) 33.3° (b) 46° (c) -66.6° (d) 240.11° (e) -369° (f) 11°
Find identities analogous to the addition identities for each expression. (a) sin(x - y) (b) cos(x - y) (c) tan(x - y)
Suppose that a tire on a truck has an outer radius of 2.5 feet. How many revolutions per minute does the tire make when the truck it travelling 60 miles per hour?
Find the angle of inclination of the following lines (see Problem 38). (a) y = √3x - 7 (b) √3x + 3y = 6
Convert the following radian measures to degree (1 radian = 180/π ≈ 57.296 degrees). (a) 3.141 (b) 6.28 (c) 5.00 (d) 0.001 (e) -0.1 (f) 36.0
Let „“1 and „“2 be two nonvertical intersecting lines with slopes m1 and m2, respectively. If Î¸, the angle from „“1 to „“2, is not a right angle,
Find the angle (in radians) from the first line to the second (see problem 40). (a) y = 2x, y = 3x (b) y = x/2, y = - x (c) 2x - 6y = 12, 2x + y = 0
Derive the formula A = ½ r2t for the area of a sector of a circle. Here r is the radius and t is the radian measure of the central angle (see Figure 17).
A regular polygon of n sides is inscribed in a circle of radius r. Find formulas for the perimeter, P, and area, A, of the polygon in terms of n and r.
An isosceles triangle is topped by a semicircle, as shown in Figure 18. Find a formula for the area A of the whole figure in terms of the side length r and angle t (radians). (We say that A is a
From a product identity, we obtain Cos x/2 cos x/4 = 1/2 [cos(3/4 x) + cos(1/4 x)] Find the corresponding sum of cosines for cos x/2 cos x/4 cos x/8 cos x/16 Do you see a generalization?
The normal high temperature for Las Vegas, Nevada, is 55°F for January 15 and 105° for July 15. Assuming that these are the extreme high and low temperatures for the year, use this information to
Tides are often measured by arbitrary height markings at some location. Suppose that a high tide occurs at noon when the water level is at 12 feet. Six hours later, a low tide with a water level of 5
Circular motion can be modeled by using the para-metric representations of the form x(t) = sin t and y(t) = cos I. (A parametric representation means that a variable, t in this case, determines both
Calculate (be sure that your calculator is in radian or degree mode as needed). (a) 56.4 tan 34.2° / sin 34.1° (b) 5.34 tan 21.3° / sin 3.1° + cot 23.5° (c) Tan 0.452 (d) Sin (-0.361)
The circular frequency v of oscillation of a point is given by v = 2π/period, What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions
We now explore the relationship between A sin(wt) + B cos(ωt) and C sin(ωt + ϕ). (a) By expanding sin(wt + ϕ) using the sum of the angles formula, show that the two expressions are equivalent if
Graph the function f(x) = sin 50 x using the window given by a y range of - 1.5 ≤ y ≤ 1.5 and the x range given by (a) [-15, 15] (b) [-10, 10] (c) [-8, 8] (d) [-1, 1] (e) [- 0.25, 0.25]
Graph the function f (x) = cos x + 1/50 sin 50x using the windows given by the following ranges of x and y. (a) -5 ≤ x ≤ 5, - 1 ≤ y ≤ 1 (b) - 1 ≤ x ≤ 1, 0.5 ≤ y ≤ 1.5 (c) - 0.1 ≤ x
Let f(x) = 3x + 2 / x2 + 1 and g(x) = 1/ 100 cos (100x). (a) Use functional composition to form h(x) = (f ͦ g) (x), as well as j(x) = (g ͦ f) (x). (b) Find the appropriate window or windows that
Suppose that a continuous function is periodic with period 1 and is linear between 0 and 025 and linear between - 0.75 and 0. In addition, it has the value 1 at 0 and 2 at 0.25. Sketch the function
Suppose that a continuous function is periodic with period 2 and is quadratic between -0.25 and 0.25 and linear between - 1.75 and - 0.25. In addition, it has the value 0 at 0 and 0.0625 at ± 0.25.
Verify the values of sin t and cos t in the table used to construct Figure 6.
Evaluate without using a calculator. (a) Tan π/6 (b) Sec π (c) Sec 3π/4 (d) Csc π/2 (e) Cot π/4 (f) Tan (-π/4)
Calculate each value for n = 1, 2, and - 22. (a) (n + 1/n)n (b) (n2 - n + 1)2 (c) 43/n (d) n√|1/n|
Suppose |x| ≤ 2. Use properties of absolute values to show that |2x2 + 3x + 2/x2 + 2| ≤ 8
Write a sentence involving the word distance to express the following algebraic sentences: (a) |x - 5| = 3 (b) |x + 1| ≤ 2 (c) |x - a| > b
Sketch the triangle with vertices A(-2, 6), B(1, 2), and C(5, 5), and show that it is a right triangle.
Find the equation of the circle with diameter AB if A = (2, 0) and B = (10, 4).
Find the center and radius of the circle with equations x2 + y2 - 8x + 6y = 0.
Find the distance between the centers of the circles with equations x2 - 2x + y2 + 2y = 2 and x2 + 6x + y2 - 4y = - 7
Find the equation of the line through the indicated point that is parallel to the indicated line, and sketch both lines. (a) (3, 2); 3x + 2y = 6 (b) (1, - 1); y = 2/3 x + 1 (c) (5,9); y = 10 (d) -3,
Write the equation of the line through (- 2, 1) that (a) Goes through (7, 3); (b) Is parallel to 3x - 2y = 5; (c) Is perpendicular to 3x + 4y = 9; (d) Is perpendicular to y = 4; (e) Has y-intercept 3.
Show that (2, - 1), (5, 3) and (11, 11) are on the same line.
In Problem 1-3, sketch the graph of each equation. 1. 3y - 4x = 6 2. x2 - 2x + y2 = 3 3. y = 2x / x2 + 2
Find the points of intersection of the graphs of y = x2 - 2x + 4 and y - x = 4.
Among all lines perpendicular to 4x - y = 2, find the equation of the one that, together with the positive x- and y-axes, forms a triangle of area 8.
For f(x) = 1/(x + 1) - 1/x, find each value (if possible). (a) f(1) (b) f(-1/2) (c) f(-1) (d) f(t - 1) (e) f(1/t)
For g(x) = (x + 1) / x, find and simplify each values. (a) g(2) (b) g(1/2) (c) g(2 + h) - g(2)/h
Which of the following function are odd? Even Neither even nor odd?(a) f(x) = 3x / x2 + 1(b) g(x) = |sin x| + cos x(c) g(x) = x3 + sin x(d)
Sketch the graph of each function.(a) f(x) = x2 - 1(b) g(x) = x / x2 + 1(c)
Suppose that f is an even function satisfying f(x) = - 1 + √x for x ≥ 0. Sketch the graph of f for - 4 ≤ x ≤ 4.
Let f(x) = x - 1 / x and g(x) = x2 + 1. Find each value. (a) (f + g) (2) (b) f ∙ g) (2) (c) (f ͦ g) (2) (d) (g ͦ f) (2) (e) f3 (- 1) (f) f2(2) + g2 (2)
Sketch the graph of each of the following, making use of translations. (a) y = ¼ x2 (b) y = ¼(x + 2)2 (c) y = - 1 + ¼ (x + 2)2
Let f(x) = √16 - x and g(x) = x4. What is the domain of each of the following? (a) f (b) f ͦ g (c) g ͦ f
Calculate each of the following without using a calclulatoe. (a) sin 570° (b) cos 9π/2 (c) cos (-13π/6)
If sin t = 0.8 and cos t < 0, find each value. (a) sin (- t) (b) cos t (c) sin 2 t (d) tan t (e) cos (π/2 - t) (f) sin (π + t)
Find the solution set, graph this set on the red line, and express this interval notation. 1. 1 - 3x > 0 2. 6x + 3 > 2x - 5 3. 3 - 2x ≤ 4x + 1 ≤ 2x + 7
Evaluate the function f(x) and g(x) from problem 9 at the following values of x: 0, 0.9, 0.999, 1.001, 1.01, 1.1, 2. (a) f(x) = x2 - 1/ x - 1 (b) g(x) = x2 - 2x + 1 / 2x2 - x - 1
Evaluate the function F(x) and G(x) from Problem 10 at the following values of x: - 1, - 0.1, - 0.01, - 0.001, 0.001, 0.01, 0.1, 1. (a) F(x) = |x| / x (b) G(x) = sin x/x
Solve the following inequalities: (a) |x - 7| < 3 (b) |x - 7| ≤ 3 (c) |x - 7| ≤ 1 (d) |x - 7| < 0.1
Solve the following inequalities: (a) |x - 2| < 1 (b) |x - 2| ≥ 1 (c) |x - 2| < 0.1 (d) |x - 2| < 0.01
In problems 1-6, find the indicated limit.1.2. 3. 4. 5. 6.
In problem 1-5, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point.1.2. 3. 4. 5.
For the function f graphed in Figure 11, find the indicated limit or function value, or state that it does not exist.(a)(b) f(-3) (c) f(-1) (d) (e) f(1) (f) (g) (h) (i) Figure 11
Follow the direction of problem 29 for the function f graphed in Figure 12.(a)(b) f(-3) (c) f(-1) (d) (e) f(1) (f) (g) (h) (i) Figure 12
For the function f graphed in Figure 13, find the indicated limit or function value, or state that it does not exist.(a) f(-3)(b) f(3)(c)(d) (e) (f) Figure 13
For the function f graphed in Figure 14, find the indicated limit or function value, or state that it does not exist.(a)(b) (c) (d) f(-1) (e) (f) f(1)
Sketch the graph of f(x) = x - [x]; then find each of the following or state that ir does not exist.(a) f(0)(b)(c) (d)
Follow the directions of Problem 35 for f(x) = x/[x].(a) f(0)(b)(c) (d)
Sketch, as best you can, the graph of a function f that satisfies all the following conditions.(a) Its domain is the interval [0, 4](b) f(0) = f(1) = f(2) = f(3) - f(4) = 1(c)(d) (e) (f)
Find each of the following limits or state that it does not exist.(a)(b) (c) (d)
Find each of the following limits or state that it does not exist.(a)(b) (c) (d)
In problem 1-5, fin the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2).1.2. 3. 4. 5.
In problem 1-6, give the appropriate Îµ-Î´ definition of each statement.1.2. 3. 4. 5. 6.
In problem 1-3, given an Îµ-Î´ proof of each limit fact.1.2.
Prove that ifThen L = M.
Let F and g be function such that 0 ‰¤ F(x) ‰¤ G(x) for all x near c, except possibly at c. Prove that ifThen
Prove thatx4 sin2 (1/x) = 0.
By considering left- and right-hand limits, prove that
Prove that if |f(x)|Then
Suppose that f (x) = L and that f(a) exists (though it may be different from L). Prove that f is bounded on some interval containing a: that is, show that there is an interval (c, d) with c
Prove that if f(x) ‰¤ g(x) for all x in some deleted interval about a and ifThen L ‰¤ M.
Suppose we wish to give an Îµ - Î´ proof thatWe begin by writing x + 6 / x4 - 4x3 + x2 + x + 6 + 1 In the form (x - 3) g(x). (a) Determine g(X). (b) Could we choose
In Problem 1-4, plot the function f(x) over interval [1.5, 2.5]. Zoom in on the graph of each function to determine how close x must be to 2 in order that f(x) is within 0.002 of 4. Your answer
In problem 1-12, use Theorem A to find each of the limits. Justify each step by appealing to a numbered statement, as in Examples 1-4.1.2. 3.
In problem 13-24, find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit.1.2. 3.
In problems 25-30, find the limits ifAnd (See Example 4). 1. 2. 3.
In problem 1-3 findFor each given function f. 1. f(x) = 3x2 2. f(x) = 3x2 + 2x + 1 3. f(x) = 1/x
Prove statement 6 of theorem A.|f(x) g(x) - 1.M = |f(x) g(x) - Lg(x) + Lg(x) - LM|= |g(X) [f(x) - L] + L[g(x) - M]|‰¤ |g(x) ||f(x) - L| + |L|| g(X) - M|Now show that ifThen there is a
Prove statement 7 of Theorem A by first giving an Îµ - Î´ proof thatAnd then applying statement 6.
Prove thatF(x) = L ‡” [f(x) - L] = 0.
Prove thatF(x) = 0 ‡” |f(x)| = 0.
Find examples to show that if
In problem 1-4, find each of the right-hand and left-hand limits or state that they do not exist.1.2. 3. 4.
Suppose that f(x) g(x) = 1 for all andProve that Does not exist.
Let R the rectangle joining the midpoint of the sides of the quadrilateral Q having vertices (± x, 0) and (0, ±1), calculate
Let y = ˆšx and consider the points M, N, O, and P with coordinates (1, 0), (0, 1), (0, 0), and (x, y) on the graph of y = ˆšx, respectively. Calculate(a)(b)
In problem 1-5, evaluate each limit.1.2. 3. 4. 5.
In problems 1-3, plot the function u(x), l(x), and f(x). Then use these graphs along with the squeeze Theorem to determine1. u(x) = |x|, l(x) = -|x|, f(x) = x sin (1/x) 2. u(x) = |x|, l(x) = - |x|,
Prove thatCos t = cos c using an argument similar to the one used in the proof that Sin t = sin c.
Prove statement 3 and 4 of Theorem A using Theorem 1.3A.Theorem 1.3AFor every real number c in the function's domain,

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