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All Textbook Solutions for Single Variable Calculus: Concepts and Contexts, Enhanced Edition
5E6E7E(a) What is wrong with the following equation? x2+x6x2=x+3 (b) In view of part (a). explain why the equation limx2x2+x6x2=limx2x+3 is correct.9EEvaluate the limit, if it exists. limx3x2+3xx2x1211E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28EIf 4x 9 f(x) x2 4x + 7 for x 0, find limx4f(x)If 2x g(x) x4 x2 + 2 for all x, evaluate limx1g(x)Prove that limx0x4cos2x=0.32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EWrite an equation that expresses the fact that a function f is continuous at the number 4.2E(a) From the graph of f , state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left. or neither.4ESketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuous but continuous from the right, at 2Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuities at 1and 4, but continuous from the left at 1 and from the right at 4Sketch the graph of a function f that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 58E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E1E2EFor the function f whose graph is given, state the following. (a) limxf(x) (b) limxf(x) (c) limx1f(x) (d) limx3f(x) (e) The equations of the asymptotesFor the function g whose graph is given, state the following. (a) limxg(x) (b) limxg(x) (c) limx0g(x) (d) limx2g(x) (e) limx2+g(x) (f) The equations of the asymptotes5E6E7E8E9ESketch the graph of an example of a function f that satisfies all of the given conditions. limx3f(x)=,limxf(x)=2,f(0)=0,fiseven11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58EA curve has equation y = f(x) (a) Write an expression for the slope of the secant line through the points P(3,f(3)) and Q(x,f(x)). (b) Write an expression for the slope of the tangent line at P.Graph the curve y = ex in the viewing rectangles [ 1, 1] by [0, 2], [0.5, 0.5] by [0.5, 1.5], and [0.1, 0.1] by [0.9, 1.1]. What do you notice about. the curve as you zoom in toward the point (0,1)?3E4EFind an equation of the tangent line to the curve at the given point. y = 4x 3x2, (2, 4)6E7E8E9E10E11E12E13EIf a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t 1.86t2. (a) Find the velocity of the rock after one second. (b) Find the velocity of the rock when t = a. (c) When will the rock hit the surface? (d) With what velocity will the rock hit the surface?The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 1/t2, where t is measured in seconds. Find the velocity of the particle at times t = a, t = 1, t = 2, and t = 3.16EFor the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning:18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EThe quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility S varies as a function of the water temperature T. (a) What is the meaning of the derivative S'(T)? What are its units? (b) Estimate the value of S'(16) and interpret it.The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon. (a) What is the meaning of the derivative S'(T)? What are its units? (b) Estimate the values of S'(15) and S'(25) and interpret them.53E54EUse the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a) f'(3) (b) f' (2) (c) f'(1) (d) f'(0) (e) f'(l) (f) f'(2) (g) f'(3)2EMatch the graph of each function in (a)(d) with the graph of its derivative in IIV. Give reasons for your choices. (a) (b) (c) (d) I II III IVTrace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 2Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 26ETrace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 2Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 2Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 2Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it. Example 1 FIGURE 1 FIGURE 211E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51EWhere is the greatest integer function f(x) = [[ x ]] not differentiable? Find a formula for[' and sketch its graph.53E54E55E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E