Calculus Volume 2
2nd Edition
ISBN: 9781630182021
Author: Gilbert Strang, Edwin Jed Herman
Publisher: OpenStax College.
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Textbook Question
Chapter 1.6, Problem 336E
In the following exercises, find each indefinite
336.
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Chapter 1 Solutions
Calculus Volume 2
Ch. 1.1 - State whether the given sums are equal or unequal....Ch. 1.1 - In the following exercises, use the rules for sums...Ch. 1.1 - In the following exercises, use the rules for sums...Ch. 1.1 - Suppose that i=1100ai=15 and i=1100bi=12 . In the...Ch. 1.1 - Suppose that i=1100ai=15 and i=1100bi=12 . In the...Ch. 1.1 - Suppose that i=1100ai=15 and i=1100bi=12 . In the...Ch. 1.1 - Suppose that i=1100ai=15 and i=1100bi=12 . In the...Ch. 1.1 - In the following exercises, use summation...Ch. 1.1 - In the following exercises, use summation...Ch. 1.1 - In the following exercises, use summation...
Ch. 1.1 - In the following exercises, use summation...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Let Ln denote the left-endpoint sum using n...Ch. 1.1 - Compute the left and right Riemann sums—L4 and R4,...Ch. 1.1 - Compute the left and right Riemann sums—L6 and R6,...Ch. 1.1 - Compute the left and right Riemann sums—L4 and R4,...Ch. 1.1 - Compute the left and right Riemann sums—L6 and R6,...Ch. 1.1 - Express the following endpoint sums in sigma...Ch. 1.1 - Express the following endpoint sums in sigma...Ch. 1.1 - Express the following endpoint sums in sigma...Ch. 1.1 - Express the following endpoint sums in sigma...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - In the following exercises, graph the function...Ch. 1.1 - Let tj denote the time that it took Tejay van...Ch. 1.1 - Let rj denote the total rainfall in Portland on...Ch. 1.1 - Let dj denote the hours of daylight and j denote...Ch. 1.1 - To help get in shape, Joe gets a new pair of...Ch. 1.1 - The following table gives approximate values of...Ch. 1.1 - The following table gives the approximate increase...Ch. 1.1 - The following table gives the approximate increase...Ch. 1.1 - The following {able gives the percent growth of...Ch. 1.1 - wIn the following exercises, estimate the areas...Ch. 1.1 - In the following exercises, estimate the areas...Ch. 1.1 - In the following exercises, estimate the areas...Ch. 1.1 - In the following exercises, estimate the areas...Ch. 1.1 - [T] Use a computer algebra system to compute the...Ch. 1.1 - [T] Use a computer algebra system to computer the...Ch. 1.1 - [T] Use a computer algebra system to compute the...Ch. 1.1 - In the following exercises, use a calculator or a...Ch. 1.1 - In the following exercises, use a calculator or a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - In the following exercises, use a calculator at a...Ch. 1.1 - For each Of the three graphs: a. Obtain a lower...Ch. 1.1 - In the previous exercise, explain why L(A) gets no...Ch. 1.1 - A unit circle is made up of n wedges equivalent to...Ch. 1.2 - In the following exercises, express the limits as...Ch. 1.2 - In the following exercises, express the limits as...Ch. 1.2 - In the following exercises, express the limits as...Ch. 1.2 - In the following exercises, express the limits as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, given Ln or Rn as...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integrals...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, evaluate the integral...Ch. 1.2 - In the following exercises, use averages of values...Ch. 1.2 - In the following exercises, use averages of values...Ch. 1.2 - In the following exercises, use averages of values...Ch. 1.2 - In the following exercises, use averages of values...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and...Ch. 1.2 - In the following exercises, use the identity...Ch. 1.2 - In the following exercises, use the identity...Ch. 1.2 - In the following exercises, use the identity...Ch. 1.2 - In the following exercises, use the identity...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, given that...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, use the comparison...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, find 1112 average...Ch. 1.2 - In the following exercises, approximate the...Ch. 1.2 - In the following exercises, approximate the...Ch. 1.2 - In the following exercises, approximate the...Ch. 1.2 - In the following exercises, approximate the...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - In the following exercises, compute the average...Ch. 1.2 - Show that the average value of sin2t over [0, 2 ]...Ch. 1.2 - Show that the average value of cos2t over [0, 2 ]...Ch. 1.2 - Explain why the graphs of a quadratic function...Ch. 1.2 - Suppose that parabola p(x)=ax2+bx+c opens downward...Ch. 1.2 - Suppose [a, b} can be subdivided into subintervals...Ch. 1.2 - Suppose f and g are continuous functions such that...Ch. 1.2 - Suppose the average value of f over [a, b] is 1...Ch. 1.2 - Suppose that [11. b] can be partitioned, taking...Ch. 1.2 - Suppose that for each i such that 1iN one has...Ch. 1.2 - Suppose that for each i such that 1iN one has...Ch. 1.2 - [T] Compute the left and right Riemann sums L10...Ch. 1.2 - [T] Compute the left and right Riemann sums, L10...Ch. 1.2 - If 151+t4dt=41.7133... , what is 151+u4du ?Ch. 1.2 - Estimate 01tdt using the left and light endpoint...Ch. 1.2 - Estimate 01tdt by comparison with the area of a...Ch. 1.2 - From the graph of sin(2(x) shown: a. Explain why...Ch. 1.2 - If f is 1-periodic (f(t+1)=f(t)) , odd, and...Ch. 1.2 - If f is 1-periodic and 01f(t)dt=A , is it...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - A Parachutist in Free Fall Figure 1.30 Skydivers...Ch. 1.3 - Consider two athletes running at variable speeds...Ch. 1.3 - Two mountain climbers start their climb at base...Ch. 1.3 - To get on a certain toll road a driver has to take...Ch. 1.3 - Set 1x(1t)dt . Find F’(2) and the average value of...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - In the following exercises, use the Fundamental...Ch. 1.3 - The graph of y=0xf(t)dt , where f is a piecewise...Ch. 1.3 - The graph of y=0xf(t)dt , where {is a piecewise...Ch. 1.3 - The graph of y=0xl(t)dt , where l is a piecewise...Ch. 1.3 - The graph of y=0xl(t)dt , where l is a piecewise...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, use a calculator to...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, evaluate each definite...Ch. 1.3 - In the following exercises, use the evaluation...Ch. 1.3 - In the following exercises, use the evaluation...Ch. 1.3 - In the following exercises, use the evaluation...Ch. 1.3 - In the following exercises, use the evaluation...Ch. 1.3 - In the following exercises, identify the mats 0f...Ch. 1.3 - In the following exercises, identify the mats 0f...Ch. 1.3 - In the following exercises, identify the mats 0f...Ch. 1.3 - In the following exercises, identify the mats 0f...Ch. 1.3 - Suppose that the number of hours of daylight en a...Ch. 1.3 - Suppose the rate of gasoline consumption in the...Ch. 1.3 - Explain why, if f is continuous aver [a, b], there...Ch. 1.3 - Explain why, if fits continuous over [a, b] and is...Ch. 1.3 - Kepler's first law states that the planets move in...Ch. 1.3 - A point on an ellipse with major axis length 2a...Ch. 1.3 - As implied earlier, according to Kepler's laws,...Ch. 1.3 - The force of gravitational attraction between the...Ch. 1.3 - The displacement from rest of a mass attached to a...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Use basic integration formulas to compute the...Ch. 1.4 - Write an integral that expresses the increase in...Ch. 1.4 - Write an integral that quantifies the change in...Ch. 1.4 - A regular N—gon (an N—sided polygon with sides...Ch. 1.4 - The area of a regular pentagon with Side length a...Ch. 1.4 - A dodecahedron is a Platonic solid with a surface...Ch. 1.4 - An icosahedron is a Platonic solid with a surface...Ch. 1.4 - Write an integral that quantifies the change in...Ch. 1.4 - Write an integral that quantifies the increase in...Ch. 1.4 - Write an integral that quantifies the increase in...Ch. 1.4 - Write an integral that quantifies the increase in...Ch. 1.4 - Suppose that a particle moves along a straight...Ch. 1.4 - Suppose that a particle moves along a straight...Ch. 1.4 - Suppose that a particle moves along a straight...Ch. 1.4 - Suppose that a particle moves along a straight...Ch. 1.4 - A ball is thrown upward from a height of 1.5 m at...Ch. 1.4 - A ball is thrown upward from a height of 3 m at an...Ch. 1.4 - The area A(t) DE a circular shape is growing at a...Ch. 1.4 - A spherical balloon is being in?ated at a constant...Ch. 1.4 - Water flows into a conical tank with...Ch. 1.4 - A horizontal cylindrical tank has cross-sectional...Ch. 1.4 - The following table lists the electrical power in...Ch. 1.4 - The average residential electrical power use (in...Ch. 1.4 - The data in the following table are used to...Ch. 1.4 - Minutes Watts Minutes Watts 15 200 165 170 30 180...Ch. 1.4 - The distribution of incomes as of 2012 in the...Ch. 1.4 - Newton’s law of gravity states that the...Ch. 1.4 - For a given motor vehicle, the maximum achievable...Ch. 1.4 - John is a 25—year 01d man who weighs 160 1b. He...Ch. 1.4 - Sandra is a 25—year old woman who weighs 120 lb....Ch. 1.4 - A motor vehicle has a maximum efficiency of 33 mpg...Ch. 1.4 - Although some engines are more efficient at given...Ch. 1.4 - [T] The following table lists the 2013 schedule of...Ch. 1.4 - [T] The following table provides hypothetical data...Ch. 1.4 - For the next two exercises use the data in the...Ch. 1.4 - For the next two exercises use the data in the...Ch. 1.4 - [T] Suppose you go on a road trip and record your...Ch. 1.4 - As a car accelerates, it does not accelerate at a...Ch. 1.4 - As a car accelerates, it does not accelerate at a...Ch. 1.4 - As a car accelerates, it does not accelerate at a...Ch. 1.4 - [T] The number 0f hamburgers 50111 at a restaurant...Ch. 1.4 - [T] An athlete runs by a motion detector, which...Ch. 1.5 - Why is u-substitution referred to as change of...Ch. 1.5 - . If f=gh , when reversing the chain rule,...Ch. 1.5 - In the following exercises, verify each identity...Ch. 1.5 - In the following exercises, verify each identity...Ch. 1.5 - In the following exercises, verify each identity...Ch. 1.5 - In the following exercises, verify each identity...Ch. 1.5 - In the following exercises, verify each identity...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following exercises, find the...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a suitable change...Ch. 1.5 - In the following Exercises, use a calculator to...Ch. 1.5 - In the following Exercises, use a calculator to...Ch. 1.5 - In the following Exercises, use a calculator to...Ch. 1.5 - In the following Exercises, use a calculator to...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - In the following exercises, evaluate the...Ch. 1.5 - If h(a)=h(b) in abg(h(x))h(x)dx , what can you say...Ch. 1.5 - Is the substitution u=1x2 02x1x2dx okay? If not,...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - In the following exercises, use a change of...Ch. 1.5 - Show that the avenge value of f(x) over an...Ch. 1.5 - €314. Find the area under the graph of f(t)=t(1 t...Ch. 1.5 - Find the area under the graph of g(t)=t(1 t 2)a...Ch. 1.5 - The area of a semicircle of radius 1 can be...Ch. 1.5 - The area of the top half of an ellipse with a...Ch. 1.5 - [T] The following graph is of a function of the...Ch. 1.5 - The following graph is of a function of the form...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, compute each...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, find each indefinite...Ch. 1.6 - In the following exercises, verify by...Ch. 1.6 - In the following exercises, verify by...Ch. 1.6 - In the following exercises, verify by...Ch. 1.6 - In the following exercises, verify by...Ch. 1.6 - Write an integral to express the area under the...Ch. 1.6 - Write an integral to express the area under the...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, use appropriate...Ch. 1.6 - In the following exercises, evaluate the definite...Ch. 1.6 - In the following exercises, evaluate the definite...Ch. 1.6 - In the following exercises, evaluate the definite...Ch. 1.6 - In the following exercises, evaluate the definite...Ch. 1.6 - In the following exercises, evaluate the definite...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, integrate using the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, does the...Ch. 1.6 - In the following exercises, f(x)0 for axb . Find...Ch. 1.6 - In the following exercises, f(x)0 for axb . Find...Ch. 1.6 - In the following exercises, f(x)0 for axb . Find...Ch. 1.6 - In the following exercises, f(x)0 for axb . Find...Ch. 1.6 - Find the area under the graph of the function...Ch. 1.6 - Compute the integral of f(x)=xex2 and find the...Ch. 1.6 - Find the limit, as N tends to in?nity, of the area...Ch. 1.6 - Show that abdtt=1/b1/adtt when 0ab .Ch. 1.6 - Suppose that f(x) > 0 for all x and that f and g...Ch. 1.6 - Use the previous exercise to find the...Ch. 1.6 - Show that if c > 0, then the integral of l/x from...Ch. 1.6 - The following exercises are intended to derive the...Ch. 1.6 - The following exercises are intended to derive the...Ch. 1.6 - The following exercises are intended to derive the...Ch. 1.6 - Pretend, fat the moment, that we do not know that...Ch. 1.6 - Pretend, fur the moment, that we do not know that...Ch. 1.6 - The sine integral, defined as S(x)=0xsinttdt is an...Ch. 1.6 - [T] The normal distribution in probability is...Ch. 1.6 - [T] Compute the right endpoint estimates R50 and...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following exercises, evaluate each integral...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - In the following Exercises, find each indefinite...Ch. 1.7 - Explain the relationship cos1t+C=dt 1 t 2 =sin1t+C...Ch. 1.7 - Explain the relationship sec1+C=dt|t| t 2...Ch. 1.7 - Explain what is wrong with the following integral:...Ch. 1.7 - Explain what is wrong with the following integral:...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following Exercises, compute the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, solve for the...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following exercises, compute each integral...Ch. 1.7 - In the following Exercises, compute each definite...Ch. 1.7 - In the following Exercises, compute each definite...Ch. 1.7 - In the following Exercises, compute each definite...Ch. 1.7 - In the following Exercises, compute each definite...Ch. 1.7 - For A > 0, compute I(A)=AAdt1+t2 and evaluate...Ch. 1.7 - For 1B , compute I(B)=1Bdtt t 2 1 and evaluate...Ch. 1.7 - Use the substitution u=2cotx and the identity...Ch. 1.7 - Approximate the points at which the graphs of...Ch. 1.7 - . [T] Approximate the points at which the graphs...Ch. 1.7 - Use the following graph to prove that...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 - Evaluate the Riemann sums L4 and R4 for the...Ch. 1 - Evaluate the Riemann sums L4 and R4 for the...Ch. 1 - Evaluate the Riemann sums L4 and R4 for the...Ch. 1 - Evaluate the Riemann sums L4 and R4 for the...Ch. 1 - Evaluate the following integrals. 447....Ch. 1 - Evaluate the following integrals. 448. 043t 1+6 t...Ch. 1 - Evaluate the following integrals. 449....Ch. 1 - Evaluate the following integrals. 450. 0/4e...Ch. 1 - Find the antiderivative. 451. dx ( x+4 )3Ch. 1 - Find the antiderivative. 452. xIn(x2)dxCh. 1 - Find the antiderivative. 453. 4x2 1 x 6 dxCh. 1 - Find the antiderivative. 454. e 2x1+e 4xdxCh. 1 - Find the derivative. 455. ddt0tsinx 1+ x 2 dxCh. 1 - Find the derivative. 456. ddx1x34t2dtCh. 1 - Find the derivative. 457. ddx1In(x)(4t+et)dtCh. 1 - Find the derivative. 458. ddx0cosxet2dtCh. 1 - The following problems consider the historic...Ch. 1 - The following problems consider the historic...Ch. 1 - The following problems consider the historic...Ch. 1 - The following problems consider the historic...Ch. 1 - The following problems consider the historic...
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- Question 2. An American option on a stock has payoff given by F = f(St) when it is exercised at time t. We know that the function f is convex. A person claims that because of convexity, it is optimal to exercise at expiration T. Do you agree with them?arrow_forwardQuestion 4. We consider a CRR model with So == 5 and up and down factors u = 1.03 and d = 0.96. We consider the interest rate r = 4% (over one period). Is this a suitable CRR model? (Explain your answer.)arrow_forwardQuestion 3. We want to price a put option with strike price K and expiration T. Two financial advisors estimate the parameters with two different statistical methods: they obtain the same return rate μ, the same volatility σ, but the first advisor has interest r₁ and the second advisor has interest rate r2 (r1>r2). They both use a CRR model with the same number of periods to price the option. Which advisor will get the larger price? (Explain your answer.)arrow_forward
- Question 5. We consider a put option with strike price K and expiration T. This option is priced using a 1-period CRR model. We consider r > 0, and σ > 0 very large. What is the approximate price of the option? In other words, what is the limit of the price of the option as σ∞. (Briefly justify your answer.)arrow_forwardQuestion 6. You collect daily data for the stock of a company Z over the past 4 months (i.e. 80 days) and calculate the log-returns (yk)/(-1. You want to build a CRR model for the evolution of the stock. The expected value and standard deviation of the log-returns are y = 0.06 and Sy 0.1. The money market interest rate is r = 0.04. Determine the risk-neutral probability of the model.arrow_forwardSeveral markets (Japan, Switzerland) introduced negative interest rates on their money market. In this problem, we will consider an annual interest rate r < 0. We consider a stock modeled by an N-period CRR model where each period is 1 year (At = 1) and the up and down factors are u and d. (a) We consider an American put option with strike price K and expiration T. Prove that if <0, the optimal strategy is to wait until expiration T to exercise.arrow_forward
- We consider an N-period CRR model where each period is 1 year (At = 1), the up factor is u = 0.1, the down factor is d = e−0.3 and r = 0. We remind you that in the CRR model, the stock price at time tn is modeled (under P) by Sta = So exp (μtn + σ√AtZn), where (Zn) is a simple symmetric random walk. (a) Find the parameters μ and σ for the CRR model described above. (b) Find P Ste So 55/50 € > 1). StN (c) Find lim P 804-N (d) Determine q. (You can use e- 1 x.) Ste (e) Find Q So (f) Find lim Q 004-N StN Soarrow_forwardIn this problem, we consider a 3-period stock market model with evolution given in Fig. 1 below. Each period corresponds to one year. The interest rate is r = 0%. 16 22 28 12 16 12 8 4 2 time Figure 1: Stock evolution for Problem 1. (a) A colleague notices that in the model above, a movement up-down leads to the same value as a movement down-up. He concludes that the model is a CRR model. Is your colleague correct? (Explain your answer.) (b) We consider a European put with strike price K = 10 and expiration T = 3 years. Find the price of this option at time 0. Provide the replicating portfolio for the first period. (c) In addition to the call above, we also consider a European call with strike price K = 10 and expiration T = 3 years. Which one has the highest price? (It is not necessary to provide the price of the call.) (d) We now assume a yearly interest rate r = 25%. We consider a Bermudan put option with strike price K = 10. It works like a standard put, but you can exercise it…arrow_forwardIn this problem, we consider a 2-period stock market model with evolution given in Fig. 1 below. Each period corresponds to one year (At = 1). The yearly interest rate is r = 1/3 = 33%. This model is a CRR model. 25 15 9 10 6 4 time Figure 1: Stock evolution for Problem 1. (a) Find the values of up and down factors u and d, and the risk-neutral probability q. (b) We consider a European put with strike price K the price of this option at time 0. == 16 and expiration T = 2 years. Find (c) Provide the number of shares of stock that the replicating portfolio contains at each pos- sible position. (d) You find this option available on the market for $2. What do you do? (Short answer.) (e) We consider an American put with strike price K = 16 and expiration T = 2 years. Find the price of this option at time 0 and describe the optimal exercising strategy. (f) We consider an American call with strike price K ○ = 16 and expiration T = 2 years. Find the price of this option at time 0 and describe…arrow_forward
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