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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions

If f sxd − 5x, show that f(x + h) f(x) 5-1 5x h h
Find the exponential function f sxd − Cbx whose graph is given. (-1,3) (1, 1) 0 x
Find the exponential function f sxd − Cbx whose graph is given. y (3,24) 0 (1,6) x
Find the domain of each function. (a) g(t)=sin(e) (b) g(t)=12'
Find the domain of each function. (a) f(x) 1- ex 1 - el-x 1+x (b) f(x) cos x e
Starting with the graph of y − ex, find the equation of the graph that results from(a) reflecting about the line y − 4(b) reflecting about the line x − 2
Starting with the graph of y − ex, write the equation of the graph that results from(a) shifting 2 units downward(b) shifting 2 units to the right(c) reflecting about the x-axis(d) reflecting about the y-axis(e) reflecting about the x-axis and then about the y-axis
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − 2s1 2 e x d
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − 1 2 12 e2x
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − e| x |
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − 222x
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − s0.5dx 2 2
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 4 and 17 and, if necessary, the transformations of Section 1.3. y − 10x12
Graph the given functions on a common screen. How are these graphs related? y − 0.9x, y − 0.6x, y − 0.3x, y − 0.1x
Graph the given functions on a common screen. How are these graphs related? y − 3x, y − 10x, y − (13)x, y − ( 1 10)
Graph the given functions on a common screen. How are these graphs related? y − ex, y − e2x, y − 8x, y − 82x
Graph the given functions on a common screen. How are these graphs related? y − 2x, y − ex, y − 5x, y − 20x
(a) How is the number e defined?(b) What is an approximate value for e?(c) What is the natural exponential function?
(a) Write an equation that defines the exponential function with base b . 0.(b) What is the domain of this function?(c) If b ± 1, what is the range of this function?(d) Sketch the general shape of the graph of the exponential function for each of the following cases.(i) b . 1 (ii) b − 1 (iii) 0
Use the Law of Exponents to rewrite and simplify the expression. x2n (a) 3n-1 x+2 (b) ab Vab
Use the Law of Exponents to rewrite and simplify the expression. (a) b8s2bd4 (b)s6y3d4 2y5
Use the Law of Exponents to rewrite and simplify the expression. (a) 84y3 (b) xs3x2d3
Use the Law of Exponents to rewrite and simplify the expression. 1 (a) 2-8 (b) x4
Suppose t is an odd function and let h − f 8 t. Is h always an odd function? What if f is odd? What if f is even?
Suppose t is an even function and let h − f 8 t. Is h always an even function?
Let f and t be linear functions with equations f sxd − m1x 1 b1 and tsxd − m2x 1 b2. Is f 8 t also a linear function? If so, what is the slope of its graph?
If you invest x dollars at 4% interest compounded annually, then the amount Asxd of the investment after one year is Asxd − 1.04x. Find A 8 A, A 8 A 8 A, and A 8 A 8 A 8 A. What do these compositions represent? Find a formula for the composition of n copies of A.
Matrix-digesting enzymes are sometimes produced by cancer cells to digest the tissue surrounding the tumor, allowing it to grow and spread. In solid tumors, the enzymes are produced only by the cells on the surface of the tumor. Suppose the diameter of a spherical tumor is growing at a rate of t
Bioavailability is a term that refers to the fraction of an antibiotic dose taken orally that is absorbed into the bloodstream.Suppose that, for a dosage of x mg, the bioavailability is hsxd − 12 x mg. If x mg enters the bloodstream, suppose that the amount eventually absorbed into the site of an
A ship is moving at a speed of 30 kmyh parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon.(a) Express the distance s between the lighthouse and the ship as a function ofd, the distance the ship has traveled since noon; that is, find f so that s − f
A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cmys.(a) Express the radius r of the balloon as a function of the time t (in seconds).(b) If V is the volume of the balloon as a function of the radius, find V 8 r and interpret it.
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys.(a) Express the radius r of this circle as a function of the time t (in seconds).(b) If A is the area of this circle as a function of the radius, find A 8 r and interpret it.
Use the given graphs of f and t to estimate the value of f s tsxdd for x − 25, 24, 23, . . . , 5. Use these estimates to sketch a rough graph of f 8 t. f 1 0 g 1 x
Use the given graphs of f and t to evaluate each expression, or explain why it is undefined.(a) f s ts2dd (b) ts f s0dd (c) s f 8 tds0d(d) s t 8 f ds6d (e) s t 8 tds22d (f) s f 8 f 2- g S 0 2. AR
Use the table to evaluate each expression.(a) f s ts1dd (b) ts f s1dd (c) f s f s1dd(d) ts ts1dd (e) s t 8 f ds3d (f) s f 8 tds6d x 1 2 3 3 4 5 6 f(x) 3 1 4 2 2 5 g(x) 6 3. 2 1 2 3
Express the function in the form f 8 t 8 h. Hsxd − sec4(sx )
Express the function in the form f 8 t 8 h. Hsxd − s8 2 1 |x |
Express the function in the form f 8 t 8 h. Rsxd − ssx 2 1
Express the function in the form f 8 t. ustd −tan t 1 1 tan t
Express the function in the form f 8 t. vstd − secst 2d tanst 2d
Express the function in the form f 8 t. Gsxd −Î x 1 1 x
Express the function in the form f 8 t. Fsxd − s3 x 1 1 s3 x
Express the function in the form f 8 t. Fsxd − cos2x
Express the function in the form f 8 t. Fsxd − s2x 1 x2d4
Find f 8 t 8 h.f sxd − tan x, tsxd −x x 2 1 , hsxd − s3 x
Find f 8 t 8 h.f sxd − sx 2 3 , tsxd − x2, hsxd − x3 1 2
Find f 8 t 8 h.f sxd − |x 2 4 |, tsxd − 2x, hsxd − sx
Find f 8 t 8 h.f sxd − 3x 2 2, tsxd − sin x, hsxd − x2
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd −x 1 1 x, tsxd − sin 2x
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd − x 1 1x, tsxd −x 1 1 x 1 2
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd − sx , tsxd − s3 1 2 x
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd − 1 2 3x, tsxd − cos x
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd − x 2 2, tsxd − x2 1 3x 1 4
Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and(d) t 8 t and their domains. f sxd − x2 2 1, tsxd − 2x 1 1
Find (a) f 1 t, (b) f 2 t, (c) f t, and (d) fyt and state their domains. f sxd − s3 2 x , tsxd − sx2 2 1
Find (a) f 1 t, (b) f 2 t, (c) f t, and (d) fyt and state their domains. f sxd − x3 1 2x2, tsxd − 3x2 2 1
Cyclic neutropenia is a blood disorder in humans characterized by periodic fluctuations in the density of a certain kind of blood cell called neutrophils. The density of neutrophils reaches highs of around 2000 cellsymL of blood and lows near zero. The period of fluctuations is approximately three
Gene frequency The frequency of a certain gene in a parasite population undergoes sinusoidal cycles as a result of coevolution with its host. It reaches a maximum frequency of 80% and a minimum of 20%, with a complete cycle taking three years. Find a function that describes the gene frequency
Volume of air in lungs In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residual volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds.Find a model
Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am.
A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude)of the star is 4.0, and its brightness varies by 60.35 magnitude. Find a
The city of New Orleans is located at latitude 308N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 am and sets at 6:18 pm in New
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 1 2 2x 2 x2
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − |x | 2 2
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.y −1 4 tanSx 2 4 D
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 12 s1 2 cos xd
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 1 2 2sx 1 3
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 2x3
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y −2 x2 2
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − sins1 2 xd
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 4 sin 3x
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − sx 2 2 2 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − x2 1 6x 1 4
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − 2s3 x
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y − sx 2 1d3
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y −1 x 1 2
(a) How is the graph of y − 2 sin x related to the graph of y − sin x? Use your answer and Figure 6 to sketch the graph of y − 2 sin x.(b) How is the graph of y − 1 1 sx related to the graph of y − sx ? Use your answer and Figure 4(a) to sketch the graph of y − 1 1 sx .
The graph of f is given. Use it to graph the following functions.(a) y − f s2xd (b) y − f s1 2xd(c) y − f s2xd (d) y − 2f s2xd y 1 0 1 1
The graph of f is given. Draw the graphs of the following functions.(a) y − f sxd 2 2 (b) y − f sx 2 2d(c) y − 22f sxd (d) y − f s1 3 xd 1 1 2 0 1 X
The graph of y − f sxd is given. Match each equation with its graph and give reasons for your choices.(a) y − f sx 2 4d (b) y − f sxd 1 3(c) y − 13 f sxd (d) y − 2f sx 1 4d(e) y − 2f sx 1 6d 9- -3 3 0 + 3 9 A
Explain how each graph is obtained from the graph of y − f sxd.(a) y − f sxd 1 8 (b) y − f sx 1 8d(c) y − 8f sxd (d) y − f s8xd(e) y − 2f sxd 2 1 (f) y − 8f s1 8 xd
Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.(a) Shift 3 units upward. (b) Shift 3 units downward.(c) Shift 3 units to the right. (d) Shift 3 units to the left.(e) Reflect about the x-axis. (f) Reflect about the y-axis.(g) Stretch
Fish growth The table gives the lengths of rock bass caught at different ages, as determined by their otoliths (ear bones in their heads). Scientists have proposed a cubic polynomial to model the data.(a) Use a cubic polynomial to model the data. Graph the polynomial together with a scatter plot of
The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods T(time of revolution in years).(a) Fit a power model to the data.(b) Kepler’s Third Law of Planetary Motion states that “The
Species–area relation for reptiles The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles.(a) Use a power function to model N as a function of A.(b) The Caribbean island of Dominica has area 291 mi2.How many
Wingspan and weight The weight W (in pounds) of a bird (that can fly) has been related to the wingspan L (in inches) of the bird by the power function L − 30.6W0.3952.(In Exercise 1.5.66 this model will be derived from data.)(a) The bald eagle has a wingspan of about 90 inches. Use the model to
Many physical quantities are connected by inverse square laws, that is, by power functions of the form f sxd − kx22.In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room
A sbestos and lung tumors When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists.(a) Find the regression line for the data.(b) Make a scatter plot and graph the regression line. Does the
Femur length Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur
Cricket chirping rate In Exercise 15 we modeled temperature as a linear function of the chirping rate of crickets from limited data. Here we use more extensive data in the following table to construct a linear model.(a) Make a scatter plot of the data.(b) Find and graph the regression line.(c) Use
P eptic ulcer rate The table on page 30 shows (lifetime)peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey.(a) Make a scatter plot of these data and decide whether a linear model is appropriate.(b) Find and graph a linear model
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. (a) y (b) y 0 x 0 x
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. (a) (b) y) x
The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi.(a) Express the monthly cost C as a function of the distance drivend, assuming that a linear relationship gives a suitable
Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 708F and 173 chirps per minute at 808F.(a) Find a linear equation that models the temperature
A bsorbing cerebrospinal fluid Cerebrospinal fluid is continually produced and reabsorbed by the body at a rate that depends on its current volume. A medical researcher finds that absorption occurs at a rate of 0.35 mLymin when the volume of fluid is 150 mL and at a rate of 0.14 mLymin when the
The relationship between the Fahrenheit sFd and Celsius sCd temperature scales is given by the linear function F − 95 C 1 32.(a) Sketch a graph of this function.(b) What is the slope of the graph and what does it represent?What is the F-intercept and what does it represent?
At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lbyin2. Below the surface, the water pressure increases by 4.34 lbyin2 for every 10 ft of descent.(a) Express the water pressure as a function of the depth below the ocean surface.(b) At what depth
Drug dosage If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of agea, pharmacists use the equation c − 0.0417Dsa 1 1d. Suppose the dosage for an adult is 200 mg.(a) Find the slope of the graph ofc. What does it represent?(b) What is
Climate change Recent studies indicate that the average surface temperature of the earth has been rising steadily.Some scientists have modeled the temperature by the linear function T − 0.02t 1 8.50, where T is temperature in °C and t represents years since 1900.(a) What do the slope and

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