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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I
Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 7. ln (1 + x) xVerify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 8. (1 + x)3 1 3xVerify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 9. 1+2x41+12xVerify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 10. ex cos x 1 + xFind the differential of each function. 11. (a) y = xe4x (b) y=1t4Find the differential of each function. 12. (a) y=1+2u1+3u (b) y = 2 sin 2Find the differential of each function. 13. (a) y=tant (b) y=1v21+v2Find the differential of each function. 14. (a) y = ln(sin ) (b) y=ex1ex(a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 15. y = ex/10, x = 0, dx = 0.1(a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 16. y=cosx,x=13,dx=0.02(a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 17. y=3+x2,x=1,dx=0.1(a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 18. y=x+1x1,x=2,dx=0.05Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 19. y = x2 4x. x = 3, x = 0.5Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 20. y = x x3, x = 0, x = 0.3Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 21. y=x2,x=3,x=0.8Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 22. y = ex, x = 0, x = 0.5Use a linear approximation (or differentials) to estimate the given number. 23. (1.999)4Use a linear approximation (or differentials) to estimate the given number. 24. 1/4.002Use a linear approximation (or differentials) to estimate the given number. 25. 10013Use a linear approximation (or differentials) to estimate the given number. 26. 100.5Use a linear approximation (or differentials) to estimate the given number. 27. e0.128EExplain, in terms of linear approximations or differentials, why the approximation is reasonable. 29. sec 0.08 130EExplain, in terms of linear approximations or differentials, why the approximation is reasonable. 31. 19.980.100232EThe edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube.The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error?35EUse differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.37E38EIf a current I passes through a resistor with resistance R, Ohms Law states that the voltage drop is V = RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the same (in magnitude) as the relative error in R.When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: F = kR4 (This is known as Poiseuilles Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood?41E42ESuppose that the only information we have about a function f is that f(1) = 5 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f(0.9) and f(1.1). (b) Are your estimates in part (a) too large or too small? Explain.44E1E2EFind the numerical value of each expression. 3. (a) cosh(ln 5) (b) cosh 54E5E6E7E8E9E10E11E12EProve the identity. 13. coth2x 1 = csch2x14E15E16E17E18E19E20E21E22EUse 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+xUcothx (h) limx0xUcothx (i) limxcschx (j) limxsinhxex24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EShow that ddx arctan(tanh x) = sech 2x.48E49EA flexible cable always hangs in the shape of a catenary y = c + a cosh(x/a), where c and a are constants and a 0 (see Figure 4 and Exercise 52). Graph several members of the family of functions y = a cosh(x/a). How does the graph change as a varies?51E52E53E54E55E56E57E58EState each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule2RCC3RCC4RCCGive several examples of how the derivative can be interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences.6RCC7RCC1RQ2RQDetermine 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. If f and g are differentiable, then ddx[f(g(x))]=f(x)g(x)4RQ5RQ6RQDetermine 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. ddx(10x)=x10x18RQ9RQ10RQ11RQDetermine 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. If f(x) = (x6 x4)5, then f(31)(x) = 0.13RQ14RQDetermine 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. If g(x) = x5, then g(x)=x5,thenlimx2g(x)g(2)x2=801RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54REUse mathematical induction (page 72) to show that if f(x) = xex, then f(n)(x) = (x + n)ex.56RE57RE58RE59RE60RE61RE62RE63RE(a) If f(x) = 4x tan x, /2 x /2, find f and f. (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f, and f.65RE66RE67RE68RE69REIf f and g are the functions whose graphs are shown, let P(x) = f(x)g(x), Q(x) = f(x)/g(x), and C(x) = f(g(x)). Find (a) P(2), (b) Q(2), and (c) C(2).71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84REFind a parabola y = ax2 + bx + c that passes through the point (1, 4) and whose tangent lines at x = 1 and x = 5 have slopes 6 and 2, respectively.86RE87RE88REA particle moves on a vertical line so that its coordinate at time t is y = t3 12t + 3, t 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval 0 1 3. (d) Graph the position, velocity, and acceleration functions for 0 t. 3. (e) When is the particle speeding up? When is it slowing down?The volume of a right circular cone is V=13r2h, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant.91RE92RE93RE94RE95RE96RE97RE98REA balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?100REThe angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is /6?102RE103RE104RE105RE106REExpress the limit as a derivative and evaluate. 107. limh016+h42h108RE109RE110RE111REShow that the length of the portion of any tangent line to the astroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.1P2P3P4P5PFind the values of the constants a and b such that limx0ax+b23x=512Show that sin-1(tanh x) = tan1(sinh x).A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (sec the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100m cast and 50 m north of the origin. At what point on the highway will the cars headlights illuminate the statue?9P10P11PFind all values of r such that the parabolas y = 4x2 and x = c + 2y2 intersect each other at right angles.13P14PThe figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod. d/ dt, in radians per second, when = /3. (b) Express the distance x = | OP | in tenus of . (c) Find an expression for the velocity of the pin P in terms of .16P17P18P19P20P21PLet P(x1, y1) be a point on the parabola y2 = 4px with focus F(p, 0). Let be the angle between the parabola and the line segment FP, and let be the angle between the horizontal line y = y1, and the parabola as in the figure. Prove that = . (Thus. by a principle of geometrical optics. light from a source placed at F will be reflected along a line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.)23P24P25P27P28P29P30PFind the two points on the curve y = x4 2x2 x that have a common tangent line.32PA lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 2/5 intersects some of these circles.34P35PExplain the difference between an absolute minimum and a local minimum.2E3E4EUse the graph to state the absolute and local maximum and minimum values of the function.Use the graph to state the absolute and local maximum and minimum values of the function.Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 4, absolute minimum at 5, local maximum at 2, local minimum at 39E10E(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.(a) Sketch the graph of a function on [1, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [1, 2] that has a local maximum but no absolute maximum.(a) Sketch the graph of a function on [1, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f(x)=12(3x1),x316E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E