Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 2.1, Problem 50E
The table shows values of the viral load
| t | 4 | 8 | 11 | 15 | 22 |
|
|
53 | 18 | 9.4 | 5.2 | 3.6 |
(a) Find the average rate of change of V with respect to t over each time interval:
(i) [4, 11] (ii) [8, 11]
(iii) [11, 15] (iv) [11, 22]
What are the units?
(b) Estimate and interpret the value of the derivative
Source: Adapted from D. Ho et al., “Rapid Turnover of Plasma Virions and CD4 Lymphocytes in HIV-1 Infection,” Nature 373 (1995); 123–26.
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Enhanced Feedback
Please try again. When calculating the average velocity, you are actually calculating the slope of the secant line between
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CE
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Chapter 2 Solutions
Calculus (MindTap Course List)
Ch. 2.1 - A curve has equation y=f(x). a Write an expression...Ch. 2.1 - Graph the curve y=sinx in the viewing rectangles...Ch. 2.1 - a Find the slope of the tangent line to the...Ch. 2.1 - a Find the slope of the tangent line to the curve...Ch. 2.1 - Prob. 5ECh. 2.1 - Prob. 6ECh. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - a Find the slope of the tangent to the curve...Ch. 2.1 - a Find the slope of the tangent to the curve y=1/x...
Ch. 2.1 - a A particle starts by moving to the right along a...Ch. 2.1 - Shown are graphs of the position functions of two...Ch. 2.1 - If a ball is thrown into the air with a velocity...Ch. 2.1 - If a rock is thrown upward on the planet Mars with...Ch. 2.1 - Prob. 15ECh. 2.1 - The displacement in feet of a particle moving in a...Ch. 2.1 - Prob. 17ECh. 2.1 - The graph of a function f is shown. a Find the...Ch. 2.1 - Prob. 19ECh. 2.1 - Find an equation of the tangent line to the graph...Ch. 2.1 - If an equation of the tangent line to the curve...Ch. 2.1 - If the tangent line to y=f(x) at 4, 3 passes...Ch. 2.1 - Sketch the graph of a function f for which f(0)=0,...Ch. 2.1 - Sketch the graph of a function g for which...Ch. 2.1 - Sketch the graph of a function g that is...Ch. 2.1 - Prob. 26ECh. 2.1 - Prob. 27ECh. 2.1 - Prob. 28ECh. 2.1 - Prob. 29ECh. 2.1 - Prob. 30ECh. 2.1 - Prob. 31ECh. 2.1 - Prob. 32ECh. 2.1 - Find f(a). f(x)=2t+1t+3Ch. 2.1 - Prob. 34ECh. 2.1 - Find f(a). f(x)=12xCh. 2.1 - Prob. 36ECh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 38ECh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - Prob. 42ECh. 2.1 - Prob. 43ECh. 2.1 - A particle moves along a straight line with...Ch. 2.1 - Prob. 45ECh. 2.1 - A roast turkey is taken from an oven when its...Ch. 2.1 - Researchers measured the average blood alcohol...Ch. 2.1 - Prob. 48ECh. 2.1 - Prob. 49ECh. 2.1 - The table shows values of the viral load V(t) in...Ch. 2.1 - The cost in dollars of producing x units of a...Ch. 2.1 - If a cylindrical tank holds 100, 000 gallons of...Ch. 2.1 - The cost of producing x ounces of gold from a new...Ch. 2.1 - The number of bacteria after t hours in a...Ch. 2.1 - Let H(t) be the daily cost in dollars to heat an...Ch. 2.1 - The quantity in pounds of a gourmet ground coffee...Ch. 2.1 - The quantity of oxygen that can dissolve in water...Ch. 2.1 - The graph shows the influence of the temperature T...Ch. 2.1 - Prob. 59ECh. 2.1 - Prob. 60ECh. 2.1 - Prob. 61ECh. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Match the graph of each function in ab with the...Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Shown is the graph of the population function P(t)...Ch. 2.2 - A rechargeable battery is plugged into a charger....Ch. 2.2 - The graph from the US Department of Energy shows...Ch. 2.2 - The graph shows how the average age M of first...Ch. 2.2 - Make a careful sketch of the graph of the sine...Ch. 2.2 - Let f(x)=x2. a Estimate the values of...Ch. 2.2 - Let f(x)=x3. a Estimate the values of...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Prob. 28ECh. 2.2 - Prob. 29ECh. 2.2 - a Sketch the graph of f(x)=6x by starting with the...Ch. 2.2 - Prob. 31ECh. 2.2 - Prob. 32ECh. 2.2 - Prob. 33ECh. 2.2 - The table gives the number N(t), measured in...Ch. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Let P represent the percentage of a citys...Ch. 2.2 - Prob. 38ECh. 2.2 - The graph of f is given. 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F(r)=5r3Ch. 2.3 - Prob. 12ECh. 2.3 - Prob. 13ECh. 2.3 - Prob. 14ECh. 2.3 - Prob. 15ECh. 2.3 - Prob. 16ECh. 2.3 - Differentiate the function. y=x2+4x+3xCh. 2.3 - Prob. 18ECh. 2.3 - Prob. 19ECh. 2.3 - Prob. 20ECh. 2.3 - Prob. 21ECh. 2.3 - Prob. 22ECh. 2.3 - Prob. 23ECh. 2.3 - Prob. 24ECh. 2.3 - Differentiate. f(x)=(5x22)(x3+3x)Ch. 2.3 - Prob. 26ECh. 2.3 - Differentiate. F(y)=(1y23y4)(y+5y3)Ch. 2.3 - Prob. 28ECh. 2.3 - Prob. 29ECh. 2.3 - Prob. 30ECh. 2.3 - Differentiate. y=x2+1x31Ch. 2.3 - Differentiate. y=1t3+2t21Ch. 2.3 - Prob. 33ECh. 2.3 - Prob. 34ECh. 2.3 - Differentiate. y=sss2Ch. 2.3 - Prob. 36ECh. 2.3 - Differentiate. f(t)=t3t3Ch. 2.3 - Prob. 38ECh. 2.3 - Differentiate. F(x)=2x5+x46xx3Ch. 2.3 - Differentiate. A(v)=v2/3(2v2+1v2)Ch. 2.3 - Differentiate. 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a Find y by implicit differentiation. b Solve the...Ch. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - Find dy/dx by implicit differentiation. x24xy+y2=4Ch. 2.6 - Find dy/dx by implicit differentiation. 2x2+xyy2=2Ch. 2.6 - Prob. 7ECh. 2.6 - Find dy/dx by implicit differentiation. x3xy2+y3=1Ch. 2.6 - Find dy/dx by implicit differentiation. x2x+y=y2+1Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Prob. 12ECh. 2.6 - Find dy/dx by implicit differentiation. x+y=x4+y4Ch. 2.6 - Prob. 14ECh. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation. xy=x2+y2Ch. 2.6 - Find dy/dx by implicit differentiation. xy=1+x2yCh. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Prob. 20ECh. 2.6 - If f(x)+x2[f(x)]3=10 and f(1)=2, find f(1).Ch. 2.6 - If g(x)+xsing(x)=x2, find g(0).Ch. 2.6 - 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A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - Graphs of the velocity functions of two particles...Ch. 2.7 - Graphs of the position functions of two particles...Ch. 2.7 - The height in meters of a projectile shot...Ch. 2.7 - If a ball is thrown vertically upward with a...Ch. 2.7 - If a rock is thrown vertically upward from the...Ch. 2.7 - Prob. 10ECh. 2.7 - a A company makes computer chips from square...Ch. 2.7 - a Sodium chlorate crystals are easy to grow in the...Ch. 2.7 - a Find the average rate of change of the area of a...Ch. 2.7 - A stone is dropped into a lake, creating a...Ch. 2.7 - A spherical balloon is being inflated. Find the...Ch. 2.7 - Prob. 16ECh. 2.7 - The mass of the part of a metal rod that lies...Ch. 2.7 - If a tank holds 5000 gallons of water, which...Ch. 2.7 - The quantity of charge Q in coulombs C that has...Ch. 2.7 - Prob. 20ECh. 2.7 - Prob. 21ECh. 2.7 - Some of the highest tides in the world occur in...Ch. 2.7 - Boyles Law states that when a sample of gas is...Ch. 2.7 - Prob. 24ECh. 2.7 - The table gives the population of the world Pt, in...Ch. 2.7 - Prob. 26ECh. 2.7 - Prob. 27ECh. 2.7 - Prob. 28ECh. 2.7 - Prob. 29ECh. 2.7 - The cost function for a certain commodity is...Ch. 2.7 - If px is the total value of the production when...Ch. 2.7 - Prob. 32ECh. 2.7 - The gas law for an ideal gas at absolute...Ch. 2.7 - Invasive species often display a wave of advance...Ch. 2.7 - In the study of ecosystems, predator-prey models...Ch. 2.7 - In a fish farm, a population of fish is introduced...Ch. 2.8 - Prob. 1ECh. 2.8 - a If A is the area of a circle with radius r and...Ch. 2.8 - Each side of a square is increasing at a rate of 6...Ch. 2.8 - Prob. 4ECh. 2.8 - A cylindrical tank with radius 5 m is being filled...Ch. 2.8 - Prob. 6ECh. 2.8 - The radius of a spherical ball is increasing at a...Ch. 2.8 - The area of a triangle with sides of lengths a and...Ch. 2.8 - Suppose y=2x+1, where x and y are functions of t....Ch. 2.8 - Prob. 10ECh. 2.8 - Prob. 11ECh. 2.8 - A particle is moving along a hyperbola xy=8. As it...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - Prob. 16ECh. 2.8 - Two cars start moving from the same point. One...Ch. 2.8 - Prob. 18ECh. 2.8 - A man starts walking north at 4 ft/s from a point...Ch. 2.8 - A baseball diamond is a square with side 90 ft. A...Ch. 2.8 - The altitude of a triangle is increasing at a rate...Ch. 2.8 - A boat is pulled into a dock by a rope attached to...Ch. 2.8 - At noon, ship A is 100 km west of ship B. Ship A...Ch. 2.8 - Prob. 24ECh. 2.8 - Water is leaking out of an inverted conical tank...Ch. 2.8 - A trough is 10 ft long and its ends have the shape...Ch. 2.8 - A water trough is 10 m long and a cross-section...Ch. 2.8 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 2.8 - Gravel is being dumped from a conveyor belt at a...Ch. 2.8 - A kite 100 ft above the ground moves horizontally...Ch. 2.8 - Prob. 31ECh. 2.8 - Prob. 32ECh. 2.8 - The top of a ladder slides down a vertical wall at...Ch. 2.8 - According to the model we used to solve Example 2,...Ch. 2.8 - If the minute hand of a clock has length r in...Ch. 2.8 - Prob. 36ECh. 2.8 - Prob. 37ECh. 2.8 - When air expands adiabatically without gaining or...Ch. 2.8 - Prob. 39ECh. 2.8 - Brain weight B as a function of body weight W in...Ch. 2.8 - Prob. 41ECh. 2.8 - Two carts, A and B, are connected by a rope 39 ft...Ch. 2.8 - Prob. 43ECh. 2.8 - Prob. 44ECh. 2.8 - A plane flies horizontally at an altitude of 5 km...Ch. 2.8 - A Ferris wheel with a radius of 10 m is rotating...Ch. 2.8 - A plane flying with a constant speed of 300 km/h...Ch. 2.8 - Two people start from the same point. One walks...Ch. 2.8 - A runner sprints around a circular track of radius...Ch. 2.8 - The minute hand on a watch is 8 mm long and the...Ch. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 2ECh. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 4ECh. 2.9 - Prob. 5ECh. 2.9 - Find the linear approximation of the function...Ch. 2.9 - Prob. 7ECh. 2.9 - Prob. 8ECh. 2.9 - 7-10 Verify the given linear approximation at a=0....Ch. 2.9 - Prob. 10ECh. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - Prob. 14ECh. 2.9 - 15-18 a Find the differential dy and b evaluate dy...Ch. 2.9 - Prob. 16ECh. 2.9 - Prob. 17ECh. 2.9 - Prob. 18ECh. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 20ECh. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 22ECh. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 24ECh. 2.9 - Prob. 25ECh. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 27ECh. 2.9 - Prob. 28ECh. 2.9 - 29-30 Explain, in terms of linear approximations...Ch. 2.9 - Prob. 30ECh. 2.9 - The edge of a cube was found to be 30 cm with a...Ch. 2.9 - The radius of a circular disk is given as 24 cm...Ch. 2.9 - The circumference of a sphere was measured to be...Ch. 2.9 - Use differentials to estimate the amount of paint...Ch. 2.9 - a Use differentials to find a formula for the...Ch. 2.9 - One side of a right triangle is known to be 20 cm...Ch. 2.9 - Prob. 37ECh. 2.9 - When blood flows along a blood vessel, the flux F...Ch. 2.9 - Establish the following rules for working with...Ch. 2.9 - Prob. 40ECh. 2.9 - Suppose that the only information we have about a...Ch. 2.9 - Suppose that we dont have a formula for g(x) but...Ch. 2.R - Write an expression for the slope of the tangent...Ch. 2.R - Prob. 2CCCh. 2.R - Prob. 3CCCh. 2.R - Define the derivative fa. Discuss two ways of...Ch. 2.R - Prob. 5CCCh. 2.R - Prob. 6CCCh. 2.R - Prob. 7CCCh. 2.R - Prob. 8CCCh. 2.R - Prob. 9CCCh. 2.R - Prob. 10CCCh. 2.R - Give several examples of how the derivative can be...Ch. 2.R - Prob. 12CCCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 2TFQCh. 2.R - Prob. 3TFQCh. 2.R - Prob. 4TFQCh. 2.R - Prob. 5TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 8TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 10TFQCh. 2.R - Prob. 11TFQCh. 2.R - Prob. 12TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 14TFQCh. 2.R - Prob. 15TFQCh. 2.R - The displacement in meters of an object moving in...Ch. 2.R - The graph of f is shown. State, with reasons, the...Ch. 2.R - Prob. 3ECh. 2.R - Prob. 4ECh. 2.R - The figure shows the graph of f,f, and f".Identify...Ch. 2.R - Prob. 6ECh. 2.R - The total cost of repaying a student loan at an...Ch. 2.R - Prob. 8ECh. 2.R - Prob. 9ECh. 2.R - Prob. 10ECh. 2.R - Prob. 11ECh. 2.R - Prob. 12ECh. 2.R - Prob. 13ECh. 2.R - Prob. 14ECh. 2.R - 1340 Calculate y. y=x2x+2xCh. 2.R - Prob. 16ECh. 2.R - Prob. 17ECh. 2.R - Prob. 18ECh. 2.R - Prob. 19ECh. 2.R - Prob. 20ECh. 2.R - Prob. 21ECh. 2.R - Prob. 22ECh. 2.R - Prob. 23ECh. 2.R - Prob. 24ECh. 2.R - Prob. 25ECh. 2.R - Prob. 26ECh. 2.R - Prob. 27ECh. 2.R - Prob. 28ECh. 2.R - 1340 Calculate y. sin(xy)=x2yCh. 2.R - Prob. 30ECh. 2.R - Prob. 31ECh. 2.R - Prob. 32ECh. 2.R - Prob. 33ECh. 2.R - Prob. 34ECh. 2.R - Prob. 35ECh. 2.R - Prob. 36ECh. 2.R - Prob. 37ECh. 2.R - Prob. 38ECh. 2.R - Prob. 39ECh. 2.R - Prob. 40ECh. 2.R - If f(t)=4t+1, find f"(2).Ch. 2.R - Prob. 42ECh. 2.R - Find y" if x6+y6=1.Ch. 2.R - Prob. 44ECh. 2.R - Prob. 45ECh. 2.R - Prob. 46ECh. 2.R - Prob. 47ECh. 2.R - Prob. 48ECh. 2.R - Prob. 49ECh. 2.R - Prob. 50ECh. 2.R - Prob. 51ECh. 2.R - Prob. 52ECh. 2.R - Prob. 53ECh. 2.R - Prob. 54ECh. 2.R - Prob. 55ECh. 2.R - How many tangent fines to the curve y=x/(x+1) pass...Ch. 2.R - If f(x)=(xa)(xb)(xc), show that...Ch. 2.R - Prob. 58ECh. 2.R - Prob. 59ECh. 2.R - If f and g are the functions whose graphs are...Ch. 2.R - Prob. 61ECh. 2.R - Prob. 62ECh. 2.R - Prob. 63ECh. 2.R - Prob. 64ECh. 2.R - Prob. 65ECh. 2.R - Prob. 66ECh. 2.R - Prob. 67ECh. 2.R - Prob. 68ECh. 2.R - Prob. 69ECh. 2.R - 6971 Find h in terms of f.and g. h(x)=f(x)g(x)Ch. 2.R - Prob. 71ECh. 2.R - Prob. 72ECh. 2.R - A particle moves on a vertical line so that its...Ch. 2.R - The volume of a right circular cone is V=13r2h,...Ch. 2.R - The mass of part of a wire is x(1+x) kilograms,...Ch. 2.R - The cost, in dollars, of producing x units of a...Ch. 2.R - Prob. 77ECh. 2.R - Prob. 78ECh. 2.R - Prob. 79ECh. 2.R - A waterskier skis over the ramp shown in the...Ch. 2.R - Prob. 81ECh. 2.R - Prob. 82ECh. 2.R - a Find the linearization of f(x)=1+3x3 at a=0....Ch. 2.R - Prob. 84ECh. 2.R - A window has the shape of a square surmounted by a...Ch. 2.R - 8688 Express the limit as a derivative and...Ch. 2.R - Prob. 87ECh. 2.R - Prob. 88ECh. 2.R - Prob. 89ECh. 2.R - Prob. 90ECh. 2.R - Prob. 91ECh. 2.R - Show that the length of the portion of any tangent...Ch. 2.P - Find points P and Q on the parabola y=1x2 so that...Ch. 2.P - Prob. 2PCh. 2.P - Show that the tangent lines to the parabola...Ch. 2.P - Prob. 4PCh. 2.P - Prob. 5PCh. 2.P - Prob. 6PCh. 2.P - Prob. 7PCh. 2.P - Prob. 8PCh. 2.P - Prob. 9PCh. 2.P - Prob. 10PCh. 2.P - How many lines are tangent to both of the circles...Ch. 2.P - If f(x)=x46+x45+21+x, calculate f(46)(3). Express...Ch. 2.P - The figure shows a rotating wheel with radius 40...Ch. 2.P - Tangent lines T1 and T2 are drawn at two points P1...Ch. 2.P - Let T and N be the tangent and normal lines to the...Ch. 2.P - Prob. 16PCh. 2.P - Prob. 17PCh. 2.P - Let P(x1,y1) be a point on the parabola y2=4px...Ch. 2.P - Suppose that we replace the parabolic mirror of...Ch. 2.P - Prob. 20PCh. 2.P - Prob. 21PCh. 2.P - Given an ellipse x2/a2+y2/b2=1, where ab, find the...Ch. 2.P - Prob. 23PCh. 2.P - Prob. 24PCh. 2.P - A lattice point in the plane is a point with...Ch. 2.P - Prob. 26PCh. 2.P - Prob. 27PCh. 2.P - Prob. 28P
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