Sketch the graphs of f(x) = sin x, g(x) = │sin x│, and h(x) = sin(│x│). In general, how are the graphs of ∣f(x)∣ and f(∣x∣) related to the graph of f ?
> One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2, 4) on the graph of f(x) = x2 (see figure). a. Find the slope of the line joining (2, 4) and (3,
> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.
> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.
> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.
> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.
> Determine whether y is a function of x. x2 y − x2 + 4y = 0
> Determine whether y is a function of x. y2 = x2 − 1
> Test for symmetry with respect to each axis and to the origin. y = x3 + x
> Sketch the graph of the function. y = -4 csc 3x
> Sketch the graph of the function. y = -sec 2πx
> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. h(x) = √x − 6
> Sketch the graph of the function. y = cot x/2
> Sketch the graph of the function. y = 1/3 tan x
> Sketch the graph of the function. y = 8 cos x/4
> Sketch the graph of the function. y = 3 sin 2x/5
> Sketch the graph of the function. y = sin πx
> Determine whether y is a function of x. x2 + y = 16
> Sketch the graph of the function. y = 9 cos x
> Solve the equation for θ, where 0 ≤ θ ≤ 2π 2 sec2 θ + tan2 θ − 5 = 0
> Evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x) = 3x – 1 f(x) – f(1) / x - 1
> Solve the equation for θ, where 0 ≤ θ ≤ 2π sec2 θ − sec θ − 2 = 0
> You drive to the beach at a rate of 120 kilometers per hour. On the return trip, you drive at a rate of 60 kilometers per hour. What is your average speed for the entire trip? Explain your reasoning.
> Solve the equation for θ, where 0 ≤ θ ≤ 2π cos3 θ = cos θ
> Solve the equation for θ, where 0 ≤ θ ≤ 2π 2 sin2 θ + 3 sin θ + 1 = 0
> solve the equation for θ, where 0 ≤ θ ≤ 2π 2 cos2 θ = 1
> solve the equation for θ, where 0 ≤ θ ≤ 2π 2 cos θ + 1 = 0
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. cos (–3π / 7)
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. sin (–π / 9)
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. csc 2π / 9
> Determine whether y is a function of x. x2 + y2 = 16
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. sec 12π / 5
> Test for symmetry with respect to each axis and to the origin. y2 = x3 − 8x
> You are in a boat 2 miles from the nearest point on the coast. You will travel to a point Q located 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time T of the trip as
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. cot 401º
> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. tan 33º
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. tan 33º
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 405º
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. -4π/3
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 13π/6
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 240º
> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. -45º
> Convert the radian measure to degree measure. -13π / 6
> Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to MathGraphs.com. x2 + y2 = 4
> Find any intercepts. y = 4x2 + 3
> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f(t) = 2 / 7 + t
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
> Use a graphing utility to compare the graph of f(x) = 4/ π (sin πx + 1/3 sin 3 πx) with the given graph. Try to improve the approximation by adding a term to f(x). Use a graphing utility to verify that your new approximat
> The monthly sales S (in thousands of units) of a seasonal product are modeled by S = 58.3 + 32.5 cos πt/6 where t is the time (in months), with t = 1 corresponding to January. Use a graphing utility to graph the model for S and determine the months when
> The model for the height h of a Ferris wheel car is h = 51 + 50 sin 8πt where t is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when t = 0. Alter the model so that the height of the car is 1 foot
> Consider an angle in standard position with r = 12 centimeters, as shown in the figure. Describe the changes in the values of x, y, sin θ, cos θ, and tan θ as θ increases continually from 0°
> How do the ranges of the cosine function and the secant function compare?
> Explain how to restrict the domain of the sine function so that it becomes a one-to-one function.
> You are given the value of tan θ. Is it possible to find the value of sec θ without finding the measure of θ? Explain.
> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. g(x) = 1 / x2 + 2
> Find any intercepts. y = 2x – 5
> Find a, b, and c such that the graph of the function matches the graph in the figure. y = a sin(bx − c)
> Find a, b, and c such that the graph of the function matches the graph in the figure. y = a cos(bx − c)
> Sketch the graph of the function. y = 1 + sin(x + π/2)
> Sketch the graph of the function. y = 1 + cos(x – π/2)
> Sketch the graph of the function. y = cos(x – π/3)
> Sketch the graph of the function. y = sin(x + π)
> Sketch the graph of the function. y = csc 2πx
> Sketch the graph of the function. y = 2 sec 2x
> Sketch the graph of the function. y = tan 2x
> Sketch the graph of the function. y = csc x/2
> Use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places. y = x5 − 5x (−0.5, y) (x, −4)
> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f (x) = x2 + 5
> Sketch the graph of the function. y = 2 tan x
> Sketch the graph of the function. y = -sin 2πx/3
> Sketch the graph of the function. y = sin x/2
> Sketch the graph of the function. y = sin x/2
> Use a graphing utility to graph each function f in the same viewing window for c = −2, c = −1, c = 1, and c = 2. Give a written description of the change in the graph caused by changing c. a. f(x) = sin x + c b. f(x) = -sin (2πx - c) c. f(x) = c cos x
> Use a graphing utility to graph each function f in the same viewing window for c = −2, c = −1, c = 1, and c = 2. Give a written description of the change in the graph caused by changing c. a. f(x) = c sin x b. f(x) = cos(cx) c. f(x) = cos(πx − c)
> Find the period of the function. y = csc 4x
> Find the period of the function. y = sec 5x
> Find the period of the function. y = 7 tan 2πx
> Use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places. y = √5 − x (2, y) (x, 3)
> Find the period of the function. y = 5 tan 2x
> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f (x) = 4 − x
> Determine the period and amplitude of each function. y = 2/3 cos πx/10
> Determine the period and amplitude of each function. y = −3 sin 4πx
> Determine the period and amplitude of each function. y = 3/2 cos x/2
> Determine the period and amplitude of each function. y = 2 sin 2x
> While traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of
> An airplane leaves the runway climbing at an angle of 18° with a speed of 275 feet per second (see figure). Find the altitude a of the plane after 1 minute.
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. cos θ/2 - cos θ = 1
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. cos2 θ + sin θ = 1
> Sketch the graph of the equation by point plotting. y = 1 / x + 2
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. sin θ = cos θ
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. sec θ csc θ = 2 csc θ
> Evaluate the function at the given value(s) of the independent variable. Then find the domain and range. a. f (−3) b. f (1) c. f (3) d. f (b2 + 1)
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. 2 cos2 θ –cos θ = 1
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. tan2 θ –tan θ = 0
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. tan2 θ = 3
> Solve the equation for θ, where 0 ≤ θ ≤ 2π. 2 sin2 θ = 1
> Find two solutions of each equation. Give your answers in radians (0 ≤ θ ≤ 2π). Do not use a calculator. a. sin θ = √3/2 b. sin θ = -√3/2
> Find two solutions of each equation. Give your answers in radians (0 ≤ θ ≤ 2π). Do not use a calculator. a. tan θ = 1 b. cot θ = -√3
> Find two solutions of each equation. Give your answers in radians (0 ≤ θ ≤ 2π). Do not use a calculator. a. sec θ = 2 b. sec θ = -2
> Sketch the graph of the equation by point plotting. y = 3/x
> Find two solutions of each equation. Give your answers in radians (0 ≤ θ ≤ 2π). Do not use a calculator. a. cos θ = √2/2 b. cos θ = -√2/2
