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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
Sketch the region that lies between the curves y − cos x and y − sin 2x and between x − 0 and x − y2. Notice that the region consists of two separate parts. Find the area of this region.
y − x cos x, y − x10 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately)the area of the region bounded by the curves.
y − x sinsx2d, y − x4 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately)the area of the region bounded by the curves.
y − cos x, y − 2 2 cos x, 0 < x < 2 Sketch the region enclosed by the given curves and find its area.
y − ex, y − xex, x − 0 Sketch the region enclosed by the given curves and find its area.
y − x2, y − 4x 2 x2 Sketch the region enclosed by the given curves and find its area.
y − 12 2 x2, y − x2 2 6 Sketch the region enclosed by the given curves and find its area.
y − x2 2 2x, y − x 1 4 Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
y − x2, y2 − x Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
y − ln x, xy − 4, x − 1, x − 3 Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
y − ex, y − x2 2 1, x − 21, x − 1 Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
Find the area of the shaded region. y=x+2 x=2 1 x y= x+1
Find the area of the shaded region. YA y=5r-r2 y=x (4,4) X
If f 9 is continuous on f0, `d and limxl` f sxd − 0, show that y`0 f 9sxd dx − 2f s0d
If f 9 is continuous on fa, bg, show that 2 yb af sxd f 9sxd dx − f f sbdg2 2 f f sadg2
Find a function f and a value of the constant a such that 2 yx af std dt − 2 sin x 2 1
If f is a continuous function such that yx 0f std dt − xe2x 1 yx 0e2t f std dt for all x, find an explicit formula for f sxd.
Niche overlap The extent to which species compete for resources is often measured by the niche overlap. If the horizontal axis represents a continuum of different resource types (for example, seed sizes for certain bird species), then a plot of the degree of preference for these resources is called
E nvironmental pollutants In Section 10.3 a model for the transport of environmental pollutants between three lakes is analyzed. It is shown that, for certain parameter values, the concentration of pollutant in one of the lakes as a function of time is given by an equation of the form xstd − k 2
A ngiotensin-converting enzyme inhibitors are medications that reduce blood pressure by dilating blood vessels.The rate of change of blood pressure with respect to dosage is given by the equation P9sdd − 2 8 lvR9sdd Rsdd3 where v is blood velocity, is blood viscosity, l is the length of the
P opulation dynamics Suppose that the birth and death rates in a population change through time according to the functions bstd and dstd. The net rate of change is defined as rstd − bstd 2 dstd.(a) Find an expression for the net change in population size between times t − a and t − b in terms
A ntibiotic pharmacokinetics An antibiotic tablet is taken and t hours later the concentration in the bloodstream is Cstd − 3se20.8t 2 e21.2td where C is measured in mgymL.(a) What is the maximum concentration of the antibiotic and when does it occur?(b) Calculate y2 0 Cstd dt and interpret your
An oil leak from a well is causing pollution at a rate of rstd − 90e20.12t gallons per month. If the leak is never fixed, what is the total amount of oil that will be spilled?
A population of honeybees increased at a rate of rstd bees per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. TA 12000 8000 4000 0 4 8 12 16 20 24 t (weeks)
Let rstd be the rate at which the world’s oil is consumed, where t is measured in years starting at t − 0 on January 1, 2000, and rstd is measured in barrels per year. What does y15 0 rstd dt represent?
The speedometer reading v on a car was observed at one-minute intervals and recorded in the chart. Use the Midpoint Rule to estimate the distance traveled by the car. t (min) v (mi/h) t (min) v (mi/h) 0 40 12345 42 45 49 52 10 68829 56 7 57 57 55 56 54
y0 2`e22x dx Evaluate the integral or show that it is divergent.
y`0 ln x x4 dx Evaluate the integral or show that it is divergent.
y`1 1s2x 1 1d3 dx Evaluate the integral or show that it is divergent.
Use the properties of integrals to verify that 0 < y1 0x4 cos x dx < 0.2
Use Property 8 of integrals (page 338) to estimate the value of y3 1sx2 1 3 dx
y cot x s1 1 2 sin x dx Use the Table of Integrals on the Reference Pages to evaluate the integral.
y ex s1 2 e2x dx Use the Table of Integrals on the Reference Pages to evaluate the integral.
y csc5t dt Use the Table of Integrals on the Reference Pages to evaluate the integral.
tsxd − ysin x 11 2 t 2 1 1 t 4 dt Find the derivative of the function.
Fsxd − yx 0t 2 1 1 t 3 dt Find the derivative of the function.
Use a graph to give a rough estimate of the area of the region that lies under the curve y − xsx , 0 < x < 4.Then find the exact area.
y1 0ex 1 1 e2x dx Evaluate the integral.
y sec tan 1 1 sec d Evaluate the integral.
y tan21 x dx Evaluate the integral.
y es3 x dx Evaluate the integral.
y sin x cosscos xd dx Evaluate the integral.
y4 1x 3y2 ln x dx Evaluate the integral.
y4 1dt s2t 1 1d3 Evaluate the integral.
y y4 2 y4 t 4 tan t 2 1 cos t dt Evaluate the integral.
y5 0ye20.6y dy Evaluate the integral.
y5 0x x 1 10 dx Evaluate the integral.
y2 1x3 ln x dx Evaluate the integral.
y x 1 2 sx2 1 4x dx Evaluate the integral.
y2 11 2 2 3x dx Evaluate the integral.
y1 0e t dt Evaluate the integral.
y1 0sins3 td dt Evaluate the integral.
y1 0v2 cossv3d dv Evaluate the integral.
y csc2 x 1 1 cot x dx Evaluate the integral.
y1 0x x2 1 1 dx Evaluate the integral.
y1 0ss4 u 1 1d2 du Evaluate the integral.
y S1 2 x x D2 dx Evaluate the integral.
y1 0s1 2 xd9 dx
y1 0s1 2 x9 d dx Evaluate the integral.
yT 0sx4 2 8x 1 7d dx Evaluate the integral.
y2 1s8x3 1 3x2 d dx Evaluate the integral.
The following figure shows the graphs off, f 9, and yx 0 f std dt. Identify each graph, and explain your choices.8. Evaluate: y a x
(a) Write y3 0 e2xy2 dx as a limit of Riemann sums, taking the sample points to be right endpoints.(b) Use the Midpoint Rule with six subintervals to estimate the value of the integral in part (a). State your answer correct to three decimal places.(c) Use the Fundamental Theorem to evaluate y3 0
If y6 0 f sxd dx − 10 and y4 0 f sxd dx − 7, find y6 4 f sxd dx.
Express lim n l `on i−1 sin xi Dx as a definite integral on the interval f0, g and then evaluate the integral.
Evaluate y1 0sx 1 s1 2 x2 d dx by interpreting it in terms of areas.
(a) Evaluate the Riemann sum for f sxd − x2 2 x 0 < x < 2 with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.(b) Use the definition of a definite integral (with right endpoints)to calculate the value of the
Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. 2. y= f(x) 0 2 6 X
If f is continuous on fa, bg, thenDetermine 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. d dx (10)-50 f(x) dx = f(x)
If y`a f sxd dx and y`a tsxd dx are both divergent, then y`a f f sxd 1 tsxdg dx is divergent.Determine 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 y`a f sxd dx and y`a tsxd dx are both convergent, then y`a f f sxd 1 tsxdg dx is convergent.Determine 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 is a continuous, decreasing function on f1, `d and lim tl`f sxd − 0, then y`1 f sxd dx is convergent.Determine 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 is continuous, then y`2` f sxd dx − lim tl`yt 2t f sxd dx.Determine 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.
All continuous functions have derivatives.Determine 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.
All continuous functions have antiderivatives.Determine 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.
y2 0 sx 2 x3d dx represents the area under the curve y − x 2 x3 from 0 to 2.Determine 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.
y5 25 sax2 1 bx 1 cd dx − 2 y5 0sax2 1 cd dx Determine 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.
y1 21Sx5 2 6x9 1 sin x s1 1 x4 d2Ddx − 0 Determine 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 t are differentiable and f sxd > tsxd for a , x , b, then f 9sxd > t9sxd for a , x , b.Determine 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 t are continuous and f sxd > tsxd for a f(x) dx = g(x) dx
If f 9 is continuous on f1, 3g, then y3 1f 9svd dv − f s3d 2 f s1d.Determine 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 is continuous on fa, bg and f sxd > 0, thenDetermine 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. Sf (x) dx= b So f(x) dx
If f is continuous on fa, bg, thenDetermine 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. fxf (x) dx = x f f (x) dx
If f is continuous on fa, bg, thenDetermine 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. 5f(x) dx = 5f f(x) dx
If f and t are continuous on fa, bg, thenDetermine 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. "b f [f(x)g(x)] dx = ( f f (x) dx ) ( 96 f g (x) dx )
If f and t are continuous on fa, bg, thenDetermine 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. S [f(x) + g(x)] dx = f* f (x) dx + g(x) dx
Explain exactly what is meant by the statement that“differentiation and integration are inverse processes.”
Define the following improper integrals.(a) y`a f sxd dx (b) yb 2`f sxd dx (c) y`2`f sxd dx
(a) State the Substitution Rule. In practice, how do you use it?(b) State the rule for integration by parts. In practice, how do you use it?
State both parts of the Fundamental Theorem of Calculus.
(a) Explain the meaning of the indefinite integral y f sxd dx.(b) What is the connection between the definite integral yb a f sxd dx and the indefinite integral y f sxd dx?
If rstd is the rate of growth of a population at time t, where t is measured in months, what does y10 6 rstd dt represent?
(a) State the Evaluation Theorem.(b) State the Net Change Theorem.
State the Midpoint Rule.
(a) Write the definition of the definite integral of a continuous function from a to b.(b) What is the geometric interpretation of yb a f sxd dx if f sxd > 0?(c) What is the geometric interpretation of yb a f sxd dx if f sxd takes on both positive and negative values? Illustrate with a diagram.
(a) Write an expression for a Riemann sum of a function f.Explain the meaning of the notation that you use.(b) If f sxd > 0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.(c) If f sxd takes on both positive and negative values, what is the geometric interpretation
Evaluate the integral, given that y`0 e2x 2 dx − 12 s 00 35. xedx
Evaluate the integral, given that y`0 e2x 2 dx − 12 s 34. xex dx Jo
Evaluate the integral, given that y`0 e2x 2 dx − 12 s 33. 10 xP 2/23-0
For what values of p is the integralconvergent? Evaluate the integral for those values of p. xp 4x
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