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All Textbook Solutions for Calculus (MindTap Course List)
4EFind the exact value of each expression. a tan(arctan10) b arcsin(sin(5/4)6EFind the exact value of each expression. tan(sin1(23))Find the exact value of each expression. csc(arccos35)Find the exact value of each expression. cos(2sin1(513))10E11E12E13E12-14 Simplify the each expression. sin(2arccosx)15E16EProve Formula 6 for the derivatives of cos1 by the same method as for Formula 3.a Prove that sin1x+cos1x=/2 b Use part a to prove Formula 6.19EProve that ddt(sec1x)=1xx2121EFind the derivative of the function. Simplify where possible. y=tan1(x2)Find the derivative of the function. Simplify where possible. y=(tan1x)2Find the derivative of the function. Simplify where possible. g(x)=arccosxFind the derivative of the function. Simplify where possible. y=sin1(2x+1)Find the derivative of the function. Simplify where possible. R(t)=arcsin(1/t)Find the derivative of the function. Simplify where possible. y=xsin1x+1x2Find the derivative of the function. Simplify where possible. y=cos1(sin1t)Find the derivative of the function. Simplify where possible. y=xsec1(x3)Find the derivative of the function. Simplify where possible. y=arctan1x1+xFind the derivative of the function. Simplify where possible. y=arctan(cos)32EFind the derivative of the function. Simplify where possible. h(t)=cot1(t)+cot1(1/t)34EFind the derivative of the function. Simplify where possible. y=arccos(b+acosxa+bcosx),0x,ab036E37E38E39E40E41E42E43E44E45E46E47EA painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer as in the figure. How far from the wall should the observer stand to get the best view? In other words, where should the observer stand so as to maximize the angle subtended at his eye by the painting?49E50E51E52E51-54 Sketch the curve using the guidelines of Section 3.5. y=xtan1x51-54 Sketch the curve using the guidelines of Section 3.5. y=earctanxIf f(x)=arctan(cos(3arcsinx)), use the graphs of f,f, and f to estimate the x-coordinates of the maximum and minimum points and inflection points of f.56E57E58EEvaluate the integral. 1/3381+x2dxEvaluate the integral. 1/21/261p2dpEvaluate the integral. 01/2sin1x1x2dxEvaluate the integral. 03/4dx1+16x2Evaluate the integral. 1+x1+x2dxEvaluate the integral. 0/2sinx1+cos2xdxEvaluate the integral. dx1x2sin1xEvaluate the integral. 1xx24dxEvaluate the integral. t21t6dt68E69E70E71E72E73EProve that, for xy1,arctanx+arctany=arctanx+y1xy if the left side lies between /2 and /275E76E77EProve the identity arcsinx1x+1=2arctanx279ELet f(x)=x arctan (1/x) if x0 and f(0)=0. a Is f continuous at 0? b Is f differentiable at 0?1-6 Find the numerical value of each expression. a sinh 0b cosh 01-6 Find the numerical value of each expression. a tanh 0b tanh 11-6 Find the numerical value of each expression. a cosh(ln5)b cosh54E1-6 Find the numerical value of each expression. a sech0b cosh116E7E8E9E10E11E12E13E7-19 Prove the identity tanh(x+y)=tanx+tanhy1+tanhxtanhy15E16E17E18E19E20E21Ea Use the graphs of sinh, cosh, and tanh in Figures 1-3 to draw the graphs of csch, sech, and coth. b Check the graphs that you sketched in part a by using a graphing device to produce them.Use the definitions of the hyperbolic functions to find each of the following limits. a limxtanhx b limxtanhx c limxsinhx d limxsinhx e limxsechx f limxcothx g limx0+cothx h limx0cothx i limxcschx j limxsinhxexProve the formulas given in Table 1 for the derivatives of the functions a cosh, b tanh, c csch, d sech, and e coth.25E26E27E28EProve the formulas given in Table 6 for the derivatives of the following functions. a cosh1 b tanh1 c csch1 d sech1 e coth130E30-45 Find the derivative. Simplify where possible. f(x)=tanhx32E30-45 Find the derivative. Simplify where possible. h(x)=sinh(x2)34E30-45 Find the derivative. Simplify where possible. G(t)=sinh(lnt)30-45 Find the derivative. Simplify where possible. y=sechx(1+lnsechx)30-45 Find the derivative. Simplify where possible. y=ecosh3x38E30-45 Find the derivative. Simplify where possible. g(t)=tcotht2+140E30-45 Find the derivative. Simplify where possible. y=cosh1x42E30-45 Find the derivative. Simplify where possible. y=xsinh1(x/3)9+x244E30-45 Find the derivative. Simplify where possible. y=coth1(secx)46E47EThe Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y=211.4920.96cosh0.03291765x for the central curve of the arch, where x and y are measured in meters and |x|91.20. a Graph the central curve. b What is the height of the arch at its center? c At what points is the height 100 m? d What is the slope of the arch at the points in part c?49EA flexible cable always hangs in the shape of a catenary y=c+acosh(x/a), where c and a are constants and a0 see Figure 4 and Exercise 52. Graph several members of the family of functions y=acosh(x/a). How does the graph change as a varies? Figure 4 Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve y=f(x) that satisfies the differential equation d2dx2=gT1+(dydx)2 where is the linear density of the cable, g is the acceleration due to gravity, T is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function y=f(x)=Tgcosh(gxT) is a solution of this differential equation.A telephone line hangs between two poles 14 m apart in the shape of the catenary s, where x and y are measured in meters. a Find the slope of this curve where it meets the right pole. b Find the angle between the line and the pole.Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve y=f(x) that satisfies the differential equation d2dx2=gT1+(dydx)2 where is the linear density of the cable, g is the acceleration due to gravity, T is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function y=f(x)=Tgcosh(gxT) is a solution of this differential equation.53E54Ea Show that any function of the form y=Asinhmx+Bcoshmx Satisfies the differential equation y=m2y. b Find y=y(x) such that y=9y,y(0)=4, and y(0)=6.56EAt what point of the curve y=coshx does the tangent have slope 1?58E59E60E59-67 Evaluate the integral. sinhxxdx62E59-67 Evaluate the integral. coshxcosh2x1dx64E59-67 Evaluate the integral. 461t29dt66E59-67 Evaluate the integral. ex1e2xdx68E69E70EShow that if a0 and b0, then there exist numbers and such that aex+bex equals either cosh(x+). In other words, almost every function of the form f(x)=aex+bex is a shifted and stretched hyperbolic sine or cosine function.1-4 Given that limxaf(x)=0limxag(x)=0limxah(x)=0limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible a limxaf(x)g(x) b limxaf(x)p(x) c limxah(x)p(x) d limxap(x)f(x) e limxap(x)q(x)2E1-4 Given that limxaf(x)=0limxag(x)=0limxah(x)=0limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible a limxa[f(x)p(x)] b limxa[p(x)q(x)] c limxa[p(x)+q(x)]4E5-6 Use the graphs of f and g and their tangent lines at 2, 0 to find limx2f(x)g(x).5-6 Use the graphs of f and g and their tangent lines at 2, 0 too find limx2f(x)g(x).The graph of a function f and its tangent line at 0 are shown. What is the value of limx0f(x)ex1?8E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx4x22x8x410E11E12E13E14E15E16E17E18E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxlnxx8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxx+x212x221E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxlnxx223E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limt08t5tt25E26E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx0ex1xx228E29E30E31E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx(lnx)2x33E34E35E36E37E38E39E40E41E42E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxxsin(/x)8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxxex/245E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxxln(11x)47E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxx3/2sin(1/x)49E50E51E52E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx0+(1x1ex1)54E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx(xlnx)56E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx0+xx58E59E60E61E62E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limxx1/x64E8-68 Find the limit. Use IHospitals Rule where appropriate. If there is a more elementary method, consider using it. If IHospitals Rule doesnt apply, explain why. limx0+(4x+1)cotx66E67E68E69E70E71E72E73E74E75E76E77E78E77-82 Use lHospitals Rule to help sketch the curve. Use the guidelines of Section 3.5. y=xex280E81E82E83-85 a Graph the function. b Use lHospitals Rule to explain the behavior as x0+ or as x. c Estimate the maximum and minimum values and then use calculus to find the exact values. d Use a graph of f to estimate the x-coordinates of the inflection points. f(x)=xx84E85E86E87EIf an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is v=mgc(1ect/m) where g is the acceleration due to gravity and c is a positive constant. In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; c is the proportionality constant. a Calculate limtv. b For fixed t, use lHospitals Rule to calculate limc0+v. What can you conclude about the velocity of a falling object in a vaccum?89E90E91E92E93E94E95EThe figure shows a sector of a circle with central angle . Let A() be the area of the segment between the chord PR and the arc PR. Let B() be the area of the triangle PQR. Find lim0+A()/B().97ESuppose f is a positive function. If limxaf(x) and limxag(x)=, show that limxa[f(x)]g(x)=0 This shows that 0 is not an indeterminate form.99E100E101E102E103E104Ea What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? b If f is a one-to-one function, how is its inverse function f1 defined? How do you obtain the graph of f1 from the graph of f? c If f is one-to-one function and f(f1(a))0, write a formula for (f1)(a).2CCa How is the inverse sine function f(x)=sin1x defined? What are its domain and range? b How is the inverse cosine function f(x)=cos1xdefined? What are its domain and range? c How is the inverse tangent function f(x)=tan1x defined? What are its domain and range? Sketch its graph.4CC5CCa How is the number e defined? b Express e as a limit. c Why is the natural exponential function y=ex used more often in calculus than the other exponential functions y=bx? d Why is the natural logarithmic function y=lnx used more often in calculus than the other logarithmic functions y=logbx?7CC8CCState whether each of the following limit forms is indeterminate. Where possible, state the limit. a 00 b c 0 d 0 e + f g h 0 i 0 j 0 k 0 l 11TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQ10TFQ11TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The inverse function of y=e3x is y=13lnx.13TFQ14TFQ15TFQ16TFQ17TFQ18TFQ19TFQ1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E13-20 Solve the equation for x. sinx=0.3