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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Given the following information about one trigonometric function, evaluate the other five functions.cos θ = 5/13 and 0 < θ < π/2
Given the following information about one trigonometric function, evaluate the other five functions.sin θ = - 4/5 and π < θ < 3π/2
Determine whether the following statements are true and give an explanation or counterexample.a. sin (a + b) = sin a + sin b.b. The equation cos θ = 2 has multiple real solutions.c. The equation sin θ = 1/2 has exactly one solution.d. The function sin (πx/12) has a period of 12.e. Of the six
Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions. 12 2х х
Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions.
Use a right triangle to simplify the given expressions. Assume x > 0. х cos ( tan V9 - х2
Use a right triangle to simplify the given expressions. Assume x > 0. x² + 16 Vx? sin ( sec -1 sec 4
Use a right triangle to simplify the given expressions. Assume x > 0.cot (tan-1 2x)
Use a right triangle to simplify the given expressions. Assume x > 0.cos (sec-1 x)
Use a right triangle to simplify the given expressions. Assume x > 0.tan (cos-1 x)
Use a right triangle to simplify the given expressions. Assume x > 0.cos (tan-1 x)
Without using a calculator, evaluate or simplify the following expressions.tan (tan-1 1)
Without using a calculator, evaluate or simplify the following expressions.csc-1 (sec 2)
Without using a calculator, evaluate or simplify the following expressions.tan-1 (tan (3π/4))
Without using a calculator, evaluate or simplify the following expressions.tan-1 (tan (π/4))
Without using a calculator, evaluate or simplify the following expressions.csc-1 (-1)
Without using a calculator, evaluate or simplify the following expressions.sec-1 2
Without using a calculator, evaluate or simplify the following expressions.cot-1 (-1/√3)
Without using a calculator, evaluate or simplify the following expressions.tan-1 √3
Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture.tan-1 x; π/2 - cot-1 x
Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture.sin-1 x; π/2 - cos-1 x
Prove the following identities.sin-1 y + sin-1 (-y) = 0
Prove the following identities.cos-1 x + cos-1 (-x) = π
Draw a right triangle to simplify the given expressions. Assume x > 0.cos (2 sin-1 x)
Draw a right triangle to simplify the given expressions. Assume x > 0.sin (2 cos-1 x)
Draw a right triangle to simplify the given expressions. Assume x > 0.sin-1 (cos θ), for 0 ≤ θ ≤ π/2
Draw a right triangle to simplify the given expressions. Assume x > 0.sin (cos-1 (x/2))
Draw a right triangle to simplify the given expressions. Assume x > 0.cos (sin-1 (x/3))
Draw a right triangle to simplify the given expressions. Assume x > 0.cos (sin-1 x)
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos-1 (cos (7π/6))
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos (cos-1 (-1))
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.sin-1 (-1)
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos-1 (-1/2)
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos-1 2
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.sin-1 √3/2
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos-1 (-√2/2)
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.tan-1 1
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.cos-1 (-1)
Without using a calculator, evaluate the following expressions or state that the quantity is undefined.sin-1 1
Solve the following equations.tan2 2θ = 1, 0 ≤ θ < π
Solve the following equations.sin θ cos θ = 0, 0 ≤ θ < 2π
Solve the following equations.sin2 θ - 1 = 0
Solve the following equations.cos 3x = sin 3x, 0 ≤ x < 2π
Solve the following equations.sin 3x = √2/2, 0 ≤ x < 2π
Solve the following equations.√2 sin x - 1 = 0
Solve the following equations.cos2 θ = 1/2, 0 ≤ θ < 2π
Solve the following equations.sin2 θ = 1/4, 0 ≤ θ < 2π
Solve the following equations.2θ cos θ + θ = 0
Solve the following equations.tan x = 1
Find the exact value of tan (3π/8).
Find the exact value of cos (π/12).
Prove that sec (x + p) = -sec x.
Prove that sec (π/2 - θ) = csc θ.
Prove that sin θ/csc θ + cos θ/sec θ = 1.
Prove that tan2 θ + 1 = sec2 θ.
Prove that tan θ = sin θ/cos θ.
Prove that sec θ = 1/cos θ.
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.cot π
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.sec (5π/2)
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.tan 3π
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.cos (-π)
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.sin (-π/2)
Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.cos 0
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.sin (16π/3)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.cot (-17π/3)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.sec (7π/6)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.cot (-13π/3)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.tan (15π/4)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.tan (-3π/4)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.sin (2π/3)
Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.cos (2π/3)
State the domain and range of sec-1 x.
The function tan x is undefined at x = ± π/2. How does this fact appear in the graph of y = tan-1 x?
Sketch the graphs of y = cos x and y = cos-1 x on the same set of axes.
Is it true that tan (tan-1 x) = x for all x? Is it true that tan-1 (tan x) = x for all x?
Why do the values of cos-1 x lie in the interval [0, π]?
Explain why the domain of the sine function must be restricted in order to define its inverse function.
What is the domain of the secant function?
Where is the tangent function undefined?
How are the sine and cosine functions related to the other four trigonometric functions?
What are the three Pythagorean identities for the trigonometric functions?
Explain what is meant by the period of a trigonometric function. What are the periods of the six trigonometric functions?
How is the radian measure of an angle determined?
Explain how a point P(x, y) on a circle of radius r determines an angle θ and the values of the six trigonometric functions at θ.
Define the six trigonometric functions in terms of the sides of a right triangle.
Prove that (logb c) (logc b) = 1, for b > 0, c > 0, b ≠ 1, and c ≠ 1.
Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f(x) = x3 + ax. Find the inverse of the following cubics using the substitution (known as Vieta’s substitution) x = z - a/(3z). Be sure to
Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f(x) = x3 + ax. Find the inverse of the following cubics using the substitution (known as Vieta’s substitution) x = z - a/(3z). Be sure to
a. Let g(x) = 2x + 3 and h(x) = x3. Consider the composite function f(x) = g(h(x)). Find f-1 directly and then express it in terms of g-1 and h-1.b. Let g(x) = x2 + 1 and h(x) = √x. Consider the composite function f(x) = g(h(x)). Find f-1 directly and then express it in terms of g-1 and h-1.c.
Consider the quartic polynomial y = f(x) = x4 - x2.a. Graph f and estimate the largest intervals on which it is one-to- one. The goal is to find the inverse function on each of these intervals.b. Make the substitution u = x2 to solve the equation y = f(x) for x in terms of y. Be sure you have
Use the following steps to prove that logb xz = z logb x.a. Let x = bp. Solve this expression for p.b. Use property E3 for exponents to express xz in terms of b and p.c. Compute logb xz and simplify.
Modify the proof outlined in Exercise 84 and use property E2 for exponents to prove that logb (x/y) = logb x - logb y.
Use the following steps to prove that logb xy = logb x + logb y.a. Let x = bp and y = bq. Solve these expressions for p and q, respectively.b. Use property E1 for exponents to express xy in terms of b, p, and q.c. Compute logb xy and simplify.
Assume that b > 0 and b ≠ 1. Show that log1/b x = - logb x.
The velocity of a skydiver (in m/s) t seconds after jumping from a plane is v(t) = 600(1 - e-kt/60)/k, where k > 0 is a constant. The terminal velocity of the skydiver is the value that v(t) approaches as t becomes large. Graph v with k = 11 and estimate the terminal velocity.
The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h = f(t) = 64t - 16t2, where t is measured in seconds after the hit.a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?b. Find the inverse function that gives the time t at
A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function Q(t) = a(1 - e-t/c), where t is measured in seconds, and a and c > 0 are physical constants. The steady-state charge is the value that Q(t) approaches as t becomes large.a.
A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function p(t) = 150 ∙ 2t/12, where t is the number of hours after the first observation.a. Verify that p(0) = 150, as claimed.b. Show
Find all the inverses associated with the following functions and state their domains.f(x) = 2x/(x + 2)
Find all the inverses associated with the following functions and state their domains.f(x) = 2/(x2 + 2)
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