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mathematics
calculus
Questions and Answers of
Calculus
The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The railway cars travel a distance of 896.5 feet at an angle of approximately 35.4°, rising to
Describe the error. cos 60o = opp / hyp = 1 / 2
In Exercises 1-4, determine whether the statement is true or false. Justify your answer. 1. sin 60° csc 60° = 1 2. sec 30° = csc 30° 3. sin 45° + cos 45° = 1 4. cos 60° − sin 30° = 0
You are given the value of tan θ. Is it possible to find the value of sec without finding the measure of θ? Explain.
Complete the table(a) Is θ or sin θ greater for θ in the interval (0, 0.5]? (b) As θ approaches 0, how do and sin θ compare? Explain.
Complete the table(a) Discuss the behavior of the sine function for in the interval [0, 1.5]. (b) Discuss the behavior of the cosine function for in the interval [0, 1.5]. (c) Use the definitions
Use the figure above and a straightedge to approximate two solutions of each equation to the nearest tenth, where 0 ≤ t ≤ 2π. To print an enlarged copy of the graph, go to MathGraphs.com. 1. (a)
The displacement from equilibrium of an oscillating weight suspended by a spring is given by y(t) = 2 cos 6t, where y is the displacement in centimeters and t is the time in is the time in seconds
The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by y(t) = 2e−t cos 6 cos 6 t, where y is the displacement in
The table shows the average high temperatures (in degrees Fahrenheit) in Boston, Massachusetts (B), and Fairbanks, Alaska (F), for selected months in 2015.(a) Use the regression feature of a graphing
A company that produces snowboards forecasts monthly sales over the next 2 years to be S = 23.1 + 0.442t + 4.3 cos πt/6 where S is measured in thousands of units and t is the time in months, with t
An airplane, flying at an altitude of 6 miles, is on a flight path that passes directly over an observer (see figure). Let θ be the angle of elevation from the observer to the plane.
Determine whether the statement is true or false. Justify your answer. 1. Determine whether the statement is true or false. Justify your answer. In each of the four quadrants, the signs of the secant
When f (t) = sin t and g (t) = cos t, is h (t) = f (t) g (t) even, odd, or neither? Explain.
Consider an angle in standard position with r = 12 centimeters, as shown in the figure. Describe he changes in the values of x, y, sin θ, cos θ, and tan θ as
(a) Use a graphing utility to complete the table.(b) Make a conjecture about the relationship between sin θ and sin(180° θ ).
(a) Use a graphing utility to complete the table.b) Make a conjecture about the relationship between cos (3Ï/2 - θ) and -sin θ.
Use a graphing utility to graph each of the six trigonometric functions. Determine the domain, range, period, and zeros of each function. Identify, and write a short paragraph describing, any
Describe the error. Your classmate uses a calculator to evaluate tan(2) and gets a result of 0.0274224385.
Let (x1, y1) and (x2, y2) be points on the unit circle corresponding to t = t1 and t = π − t1, respectively. (a) Identify the symmetry of the points (x1, y1) and(x2, y2). (b) Make a conjecture
With a graphing utility in radian and parametric modes, enter the equations X1T = cos T and Y1T = sin T and use the settings below. Tmin = 0, Tmax = 6.3, Tstep = 0.1 Xmin = −1.5, Xmax = 1.5, Xscl =
Determine the quadrant in which lies. 1. sin > 0, cos > 0 2. sin < 0, cos < 0 3. csc > 0, tan < 0 4. sec > 0, cot < 0
Find the exact values of the remaining trigonometric functions of satisfying the given conditions. 1. tan θ = 15/8, sin θ > 0 2. cos θ = 8/17, tan θ < 0
The terminal side of lies on the given line in the specified quadrant. Find the exact values of the six trigonometric functions of by finding a point on the line. Line
Evaluate the trigonometric function of the quadrantal angle, if possible. 1. sin 0 2. csc 3π/2 3. sec 3π/2 4. sec π
Find the reference angle θ. Then sketch in standard position and label θ. 1. θ = 160° 2. θ = 309° 3. θ = −125° 4. θ = −215°
Find the exact values of the six trigonometric functions of each angle .1. (a)(b) 2. (a) (b)
evaluate the sine, cosine, and tangent of the angle without using a calculator. 1. 225° 2. 300° 3. 750° 4. 675°
use the function value to find the indicated trigonometric value in the specified quadrant. Function Value ........................ Quadrant Trigonometric ............................... Value 1.
Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is in the correct mode.) 1. sin 10° 2. tan 304° 3. cos(−110°) 4.
Find two solutions of each equation. Give your answers in degrees (0o ≤ θ < 360o) and in radians (0 ≤ θ < 2π). Do not use a calculator. 1. (a) sin θ = 1/2 (b) sin = −1/2 2. (a)
The point is on the terminal side of an angle in standard position. Find the exact values of the six trigonometric functions of the angle. 1. (5, 12) 2. (8, 15)
Find the point (x, y) on the unit circle that corresponds to the real number t. Use the result to evaluate sin t, cos t, and tan t. 1. t = π/4 2. t = π/3 3. t = 5π/6
describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. 1. f (x) = cos x g(x) = cos 5x 2. f (x) = sin x g(x) = 2 sin x 3. f (x) = cos 2x g(x) =
Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.) 1. f (x) = sin x g (x) = sin x/3 2. f (x) = sin x g (x) = 4 sinx
Sketch the graph of the function. (Include two full periods.) 1. y = 5 sin x 2. y = ¼ sin x
Find the period and amplitude. 1. y = 2 sin 5x 2. y = 3 cos 2x 3. y =3/4 cos πx/2 4. y = −5 sin πx/3
g is related to a parent function f (x) sin x or f (x) cos x. (a) Describe the sequence of transformations from f to g. (b) Sketch the graph of g. (c) Use function otation to write g in terms
use a graphing utility to graph the function. (Include two full periods.) Be sure to choose an appropriate viewing window. 1. y = -2 sin (4x + π) 2. y = -4 sin (2/3 x - π/3) 3. y = cos ( 2πx -
Find a and d for the functionf (x) = a cos x d such that the graph of f matches the figure.1.2. 3. 4.
find a, b, and c for the function f (x) a sin (bx - c) such that the graph of f matches the figure.1.2.
Use a graphing utility to graph y1 and y2 in the interval [-2π, 2π]. Use the graphs to find real numbers x such that y1 y2. 1. y1 = sin x, y2 = −1/2 2. y1 = cos x, y2 = −1
For a person exercising, the velocity v (in liters per second) of airflow during arespiratory cycle (the time from the beginning of one breath to the beginning of the next) is modeled by v = 1.75
For a person at rest, the velocity v (in liters per second) of airflow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is modeled by v = 0.85 sin
When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approximated by y = 0.001 sin 880πt, where t is the time (in seconds). (a)
The table shows the percent y (in decimal form) of the moon's face illuminated on day x in the year 2018, where x = 1 corresponds to January1.(a) Create a scatter plot of the data. (b) Find a
The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas L and International Falls I for month t, where t = 1 corresponds to January.(a) A model for the temperatures
The height h (in feet) above ground of a seat on a Ferris wheel at time t (in seconds) is modeled by h (t) = 53 + 50 sin {(π/10) t - π/2} (a) Find the period of the model. What does the period
The daily consumption C (ingallons) of diesel fuel on a farm is modeled by C = 30.3 + 21.6 sin(2πt / 365 + 10.9) where t is the time (in days), with t = 1 corresponding to January 1. (a) What is
Determine whether the statement is true or false. Justify your answer. 1. The graph of g(x) = sin(x + 2π) is a translation of the graph of f (x) = sin x exactly one period to the right, and the two
The figure below shows the graph of y = sin (x - c) for c = - Ï/4, 0, and Ï/4.(a) How does the value of c affect the graph? (b) Which graph is equivalent to that of y = -cos (x
Sketch the graph of y = cos bx for b = 1/2, 2, and 3. How does the value of b affect the graph? How many complete cycles of the graph occur between 0 and 2 for each value of b?
Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials sin x ≈ x - x3/2! + x5/5! And cos ≈ 1 - x2/2! + x4/4!, where x is in radians. (a) Use a
Use the polynomial approximations of the sine and cosine functions in Exercise 94 to approximate each function value. Compare the results with those given by a calculator. Is the error in the
1. The tangent, cotangent, and cosecant functions are ________, so the graphs of these functions have symmetry with respect to the ________. 2. The graphs of the tangent, cotangent, secant, and
In Exercises 1-4, sketch the graph of the function. (Include two full periods.) 1. y = 1 / 3 tan x 2. y = - 1 / 2 tan x 3. y = - 1 / 2 sec x 4. y = 1 / 4 sec x
In Exercises 1-4, use a graphing utility to graph the function. (Include two full periods.) 1. y = tan x / 3 2. y = - tan 2x 3. y = - 2 sec 4x 4. y = sec π x
In Exercises 1-4, find the solutions of the equation in the interval [-2π, 2π]. Use a graphing utility to verify your results. 1. tan x = 1 2. tan x = √3 3. cot x = -√3 4. cot x = 1
In Exercises 1-4, use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. 1. f (x) = sec x 2. f (x) = tan x 3. g(x) = cot x 4.
In Exercises 1-4, match the function with its graph. Describe the behavior of the function as x approaches zero. [The graphs are labeled (a), (b), (c), and (d).]a.b. c. d. 1. f (x) = x
In Exercises 1-4, graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions. 1. f (x) = sin x + cos(x + π / 2), g (x) = 0 2. f (x) = sin x -
In Exercises 1-4, use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as x increases without
In Exercises 1-4, use a graphing utility to graph the function. Describe the behavior of the function as x approaches zero. 1. y = 6 / x + cos x, x > 0 2. y = 4 /x + sin 2x, x > 0 3. g (x) = sin x
The normal monthly high temperatures H (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by H(t) = 57.54 18.53 cos Ït / 6 - 14.03 sin Ït / 6 and the
The projected monthly sales S (in thousands of units) of lawn mowers are modeled by S = 74 + 3t - 40 cos πt / 6 where t is the time (in months), with t = 1 corresponding to January. (a) Graph the
A television camera is on a reviewing platform 27meters from the street on which a parade asses from left to right (see figure). Write the distance d from the camera to a unit in the parade as
A plane flying at an altitude of 7 miles above a radar antenna passes directly over the radar antenna (see figure). Let d be the ground distance from the antenna to the point directly under the plane
In Exercises 1 and 2, determine whether the statement is true or false. Justify your answer. 1. You can obtain the graph of y = csc x on a calculator by graphing the reciprocal of y = sin x. 2. You
Consider the function f (x) = x − cos x. (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and 1. Use the graph to approximate the zero. (b) Starting
In Exercises 1-4, match the function with its graph. State the period of the function. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]a.b. c. d. e. f. 1. y = sec 2x 2. y = tan x / 2 3. y =
In Exercises 1 and 2, use a graphing utility to graph the function. Use the graph to determine the behavior of the function as x → c. (Note: The notation x → c+ indicates that x approaches c
In Exercises 1 and 2, use a graphing utility to graph the function. Use the graph to determine the behavior of the function as x → c. a. x → (π / 2) + b. x → (π / 2) - c. x → (- π / 2)
Function Alternative Notation Domain Range 1. y = arcsin x ___________ ______ - π / 2 ≤ y ≤ π / 2 2. _______ y = cos 1 x - 1 ≤ x ≤ 1 ______ 3. y = arctan x
A television camera at ground level films the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height
Different types of granular substances naturally settle at different angles when stored in cone-shaped piles. This angle θ is called the angle of repose (see figure). When rock salt is
When shelled corn is stored in a cone-shaped pile 20 feet high, the diameter of the pile's base is about 94 feet. (a) Draw a diagram that gives a visual representation of the problem. Label the known
A photographer takes a picture of a three-foot-tall painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle β subtended
An airplane flies at an altitude of 6 miles toward a point directly over an observer. Consider θ and x as shown in the figure.(a) Write θ as a function of x. (b) Find
A police car with its spotlight on is parked 20 meters from a warehouse. Consider and x as shown in the figure.(a) Write θ as a function of x. (b) Find θ when x = 5 meters
In Exercises 1-4, determine whether the statement is true or false. Justify your answer 1. sin 5π / 6 = 1 / 2 → arcsin 1 / 2 = 5π / 6 2. tan (- π / 4) = - 1 → arctan (-1) = - π / 4 3.
Define the inverse cotangent function by restricting the domain of the cotangent function to the interval (0,π), and sketch the graph of the inverse trigonometric function.
Define the inverse secant function by restricting the domain of the secant function to the intervals [0, π/2) and ( π/2, π], and sketch the graph of the inverse trigonometric function.
Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals [−π/2, 0) and (0, π/2], and sketch the graph of the inverse trigonometric function.
Use the results of Exercises 111–113 to explain how to graph(a) The inverse cotangent function,(b) The inverse secant function, and(c) The inverse co-secant function on a graphing utility.
In Exercises 1-4, use the results of following Exercises to find the exact value of the expression. 1. arcsec √2 2. arcsec 1 3. arccot (-1) 4. arccot (- √3)
In Exercises 1-4, use the results of following Exercises and a calculator to approximate the value of the expression. Round your result to two decimal places. 1. arcsec 2.54 2. arcsec (−1.52) 3.
In calculus, it is shown that the area of the region bounded by the graphs of y = 0, y = 1 / (x2 + 1), x = a, and x = b (see figure) is given by Area = arctan b arctan a.Find the area
Use a graphing utility to graph the functions f (x) = √x and g(x) = 6 arctan x. For x > 0, it appears that g > f. Explain how you know that there exists a positive real number a such that g < f for
Consider the functions f (x) = sin x and f -1 (x) = arcsin x. (a) Use a graphing utility to graph the composite functions f o f −1 and f −1 o f. (b) Explain why the graphs in part (a) are not
Prove each identity. (a) arcsin(−x) = −arcsin x (b) arctan(−x) = −arctan x (c) arctan x + arctan 1 / x = π / 2, x > 0 (d) arcsin x + arccos x = π / 2 (e) arcsin x = arctan x / √1 - x2
In Exercises 19 and 20, use a graphing utility to graph f, g, and y = x in the same viewing window to verify geometrically that g is the inverse function of f. (Be sure to restrict the domain of f
In Exercises 1-5, use a calculator to approximate the value of the expression, if possible. Round your result to two decimal places. 1. arccos 0.37 2. arcsin 0.65 3. arcsin(−0.75) 4.
In Exercises 1 and 2, determine the missing coordinates of the points on the graph of the function.1.2.
In Exercises 1-4, use an inverse trigonometric function to write as a function of x.1.2. 3. 4.
In Exercises 1-4, find the exact value of the expression, if possible. 1. sin(arcsin 0.3) 2. tan(arctan 45) 3. cos[arccos(−√3)] 4. sin[arcsin(−0.2)]
In Exercises 1-4, find the exact value of the expression, if possible. 1. arcsin 1 / 2 2. arcsin 0 3. arccos 1 / 2 4. arccos 0
In Exercises 1-4, find the exact value of the expression, if possible. 1. sin(arctan 3/4) 2. cos(arcsin 4/5) 3. cos(tan 12) 4. sin(cos 1√5)
In Exercises 1-4, write an algebraic expression that is equivalent to the given expression. 1. cos(arcsin 2x) 2. sin(arctan x) 3. cot(arctan x) 4. sec(arctan 3x)
In Exercises 73 and 74, use a graphing utility to graph f and g in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the
In Exercises 1-4, complete the equation. 1. arctan 9 / x = arcsin (__), x > 0 2. arcsin √36 - x2 / 6 = arccos (____), 0 ≤ x ≤ 6 3. arrocs 3 / √x2 - 2x + 10 = arccos(____), 4. arccos x - 2
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