Calculus Volume 1
17th Edition
ISBN: 9781938168024
Author: Strang, Gilbert
Publisher: OpenStax College
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Textbook Question
Chapter 1.1, Problem 4E
For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.
4.
| x | y | x | y |
| 1 | 1 | 5 | 1 |
| 2 | 1 | 6 | 1 |
| 3 | 1 | 7 | 1 |
| 4 | 1 |
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1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
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Chapter 1 Solutions
Calculus Volume 1
Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, (a) determine the...Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the values for...
Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the values for...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, find the domain,...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, set up a table to...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, use the vertical line...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - For the following exercises, for each pair of...Ch. 1.1 - The table below lists the NBA championship winners...Ch. 1.1 - The area A of a square depends on the length of...Ch. 1.1 - [T] The volume of a cube depends on the length of...Ch. 1.1 - [T] A rental car company rents cars for a flat fee...Ch. 1.1 - [T] A vehicle has a 20-gal tank and gets 15 mpg....Ch. 1.1 - [T] The volume V of a sphere depends on the length...Ch. 1.1 - [T] A certain bacterium grows in culture in a...Ch. 1.1 - [T] An American tourist visits Paris and must...Ch. 1.1 - [T] The manager at a skateboard shop pays his...Ch. 1.1 - Use a graphing calculator to graph the half-circle...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, for each pair of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, write the equation of...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each linear...Ch. 1.2 - For the following exercises, for each polynomial,...Ch. 1.2 - For the following exercises, for each polynomial,...Ch. 1.2 - For the following exercises, for each polynomial,...Ch. 1.2 - For the following exercises, for each polynomial,...Ch. 1.2 - For the following exercises, for each polynomial,...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, use the graph of...Ch. 1.2 - For the following exercises, for each of the...Ch. 1.2 - For the following exercises, for each of the...Ch. 1.2 - For the following exercises, for each of the...Ch. 1.2 - For the following exercises, for each of the...Ch. 1.2 - For the following exercises, determine whether the...Ch. 1.2 - For the following exercises, determine whether the...Ch. 1.2 - For the following exercises, determine whether the...Ch. 1.2 - For the following exercises, determine whether the...Ch. 1.2 - For the following exercises, determine whether the...Ch. 1.2 - [T] A company purchases some computer equipment...Ch. 1.2 - [T] Total online shopping during the Christmas...Ch. 1.2 - [T] A family bakery makes cupcakes and sells them...Ch. 1.2 - [T] A house purchased for $250,000 is expected to...Ch. 1.2 - [T] A car was purchased for $26,000. The value of...Ch. 1.2 - [T] A condominium in an upscale part of the city...Ch. 1.2 - [T] The total cost C (in thousands of dollars) to...Ch. 1.2 - [T] A professor asks her class to report the...Ch. 1.2 - [T] The output (as a percent of total capacity) of...Ch. 1.2 - [T] The admissions office at a public university...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - For the following exercises, convert each angle in...Ch. 1.3 - Evaluate the following functional values. 123....Ch. 1.3 - Evaluate the following functional values. 124....Ch. 1.3 - Evaluate the following functional values. 125....Ch. 1.3 - Evaluate the following functional values. 126....Ch. 1.3 - Evaluate the following functional values. 127....Ch. 1.3 - Evaluate the following functional values. 128....Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, consider triangle...Ch. 1.3 - For the following exercises, P is a point on the...Ch. 1.3 - For the following exercises, P is a point on the...Ch. 1.3 - For the following exercises, P is a point on the...Ch. 1.3 - For the following exercises, P is a point on the...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, simplify each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, verify that each...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, solve the...Ch. 1.3 - For the following exercises, each graph is of the...Ch. 1.3 - For the following exercises, each graph is of the...Ch. 1.3 - For the following exercises, each graph is of the...Ch. 1.3 - For the following exercises, each graph is of the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - For the following exercises, find a. the...Ch. 1.3 - [T] The diameter of a wheel rolling on the ground...Ch. 1.3 - [T] Find the length of the arc intercepted by...Ch. 1.3 - [T] As a point P moves around a circle, the...Ch. 1.3 - [T] A total of 250,000 m2 of land is needed to...Ch. 1.3 - [T] The area of an isosceles triangle with equal...Ch. 1.3 - [T] A panicle travels in a circular path at a...Ch. 1.3 - [T] An alternating current for outlets in a home...Ch. 1.3 - [T] The number of hours of daylight in a northeast...Ch. 1.3 - [T] Suppose that T=50+10sin[12(t8)] is a...Ch. 1.3 - [T] The function H(t)=8sin(6t) models the height H...Ch. 1.4 - Consider the graph in Figure 1.42 of the function...Ch. 1.4 - Now consider other graphs of the form...Ch. 1.4 - Repeat for A = 1, B = 2. Is there any relationship...Ch. 1.4 - Complete the following table, adding a few choices...Ch. 1.4 - 5. Try to figure out the formula for the y-values.Ch. 1.4 - 6. The formula for the x-values is a little...Ch. 1.4 - 7. If you found formulas for parts (5) and (6),...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, use the horizontal...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, a. find the inverse...Ch. 1.4 - For the following exercises, use the graph of f...Ch. 1.4 - For the following exercises, use the graph of f...Ch. 1.4 - For the following exercises, use the graph of f...Ch. 1.4 - For the following exercises, use the graph of f to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, use composition to...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - For the following exercises, evaluate the...Ch. 1.4 - The function C=T(F)=(5/9)(F32) converts degrees...Ch. 1.4 - [T] The velocity V (in centimeters per second) of...Ch. 1.4 - A function that converts dress sizes in the United...Ch. 1.4 - [T] The cost to remove a toxin from a lake is...Ch. 1.4 - [T] A race car is accelerating at a velocity given...Ch. 1.4 - [T] An airplane’s Mach number M is the ratio of...Ch. 1.4 - [T] Using =2sin1(1M) , find the Mach number M for...Ch. 1.4 - [T] The temperature (in degrees Celsius) of a city...Ch. 1.4 - [T] The depth (in feet) of water at a dock changes...Ch. 1.4 - [T] All object moving in simple harmonic motion is...Ch. 1.4 - A local art gallery has a portrait 3 ft in height...Ch. 1.4 - [T] Use a calculator to evaluate...Ch. 1.4 - [T] Use a calculator to evaluate...Ch. 1.5 - For the following exercises, evaluate the given...Ch. 1.5 - For the following exercises, evaluate the given...Ch. 1.5 - For the following exercises, evaluate the given...Ch. 1.5 - For the following exercises, evaluate the given...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, match the exponential...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, write the equation in...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, sketch the graph of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, use properties of...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the exponential...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, solve the logarithmic...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - For the following exercises, rise the...Ch. 1.5 - Rewrite the following expressions in terms of...Ch. 1.5 - [T] The number of bacteria N in a culture after t...Ch. 1.5 - [T] The demand D (in millions of barrels) for oil...Ch. 1.5 - [T] The amount A of a $100,000 investment paying...Ch. 1.5 - [T] An investment is compounded monthly,...Ch. 1.5 - [T] The concentration of hydrogen ions in a...Ch. 1.5 - [T] Iodine-131 is a radioactive substance that...Ch. 1.5 - [T] According to the World Bank, at the end of...Ch. 1.5 - [T] The amount A accumulated after 1000 dollar’s...Ch. 1.5 - [T] A bacterial colony grown in a lab is known to...Ch. 1.5 - [T] The rabbit population on a game reserve...Ch. 1.5 - [T] The 1906 earthquake in Sail Francisco had a...Ch. 1 - True or False? Justify your answer with a proof or...Ch. 1 - True or False? Justify your answer with a proof or...Ch. 1 - True or False? Justify your answer with a proof or...Ch. 1 - True or False? Justify your answer with a proof or...Ch. 1 - For the following problems, state the domain and...Ch. 1 - For the following problems, state the domain and...Ch. 1 - For the following problems, state the domain and...Ch. 1 - For the following problems, state the domain and...Ch. 1 - Find the degree, y-intercept, and zeros for the...Ch. 1 - Find the degree, y-intercept, and zeros for the...Ch. 1 - Simplify the following trigonometric expressions....Ch. 1 - Simplify the following trigonometric expressions....Ch. 1 - Solve the following trigonometric equations on the...Ch. 1 - Solve the following trigonometric equations on the...Ch. 1 - Solve the following logarithmic equations. 324....Ch. 1 - wSolve the following logarithmic equations. 325....Ch. 1 - Are the following functions one-to-one over their...Ch. 1 - Are the following functions one-to-one over their...Ch. 1 - For the following problems, determine the largest...Ch. 1 - For the following problems, determine the largest...Ch. 1 - A car is racing along a circular track with...Ch. 1 - For the following problems, consider a restaurant...Ch. 1 - For the following problems, consider a restaurant...Ch. 1 - For the following problems, consider a restaurant...Ch. 1 - For the following problems, consider a restaurant...Ch. 1 - For the following problems, consider radioactive...Ch. 1 - For the following problems, consider radioactive...
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