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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
f(x) = 2x + 3 and h(x) = -3x - 12.For each of the above sums of functions, graph each component piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and plot the sum.
F(x) = x2 + 1 and G(x) = x + 1.For each of the above sums of functions, graph each component piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and plot the sum.
A scientist measures the body temperature of bandicoots every day during the winter, and does so at three different altitudes: 500 m, 750 m, and 1000 m. Identify the variables and parameters in the above situations, give the units they might be measured in, and choose an appropriate letter or
F(x) = x2 + 1 and H(x) = -x + 1.For each of the above sums of functions, graph each component piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and plot the sum.
f(x) = 2x + 3 and g(x) = 3x - 5.For each of the above products of functions, graph each component piece. Compute the value of the product at x = -2, x = -1, x = 0, x = 1, and x = 2 and graph the result.
f(x) = 2x + 3 and h(x) = -3x - 12.For each of the above products of functions, graph each component piece. Compute the value of the product at x = -2, x = -1, x = 0, x = 1, and x = 2 and graph the result.
F(x) = x2+ 1 and G(x) = x + 1.For each of the above products of functions, graph each component piece. Compute the value of the product at x = -2, x = -1, x = 0, x = 1, and x = 2 and graph the result.
F(x) = x2 + l and H(x) = -x + l.For each of the above products of functions, graph each component piece. Compute the value of the product at x = -2, x = -1, x = 0, x = 1, and x = 2 and graph the result.
f(x) = 2x + 3. Find the inverses of each of the above functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function.
g(x) = 3x - 5. Find the inverses of each of the above functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function.
G(y) = 1/(2 + y) for y ≥ 0. Find the inverses of each of the above functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function.
F(y) = y2 + 1 for y ≥ 0. Find the inverses of each of the above functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function.
f(x) = 2x + 3. Mark the point (1, f(l)) on the graphs of f and f-1. Graph each of the above functions and its inverse. Mark the given point on the graph of each function.
f(x) = x + 5 at x = 0, x = 1, and x = 4. Compute the values of the above functions at the points indicated and sketch a graph of the function.
g(x) = 3x - 5. Mark the point (1, g(l)) on the graphs of g and g-1. Graph each of the above functions and its inverse. Mark the given point on the graph of each function.
G(y) = 1/(2 + y). Mark the point (1, G(l)) on the graphs of G and G-1. Graph each of the above functions and its inverse. Mark the given point on the graph of each function.
F(y) = y2 + 1 for y ≥ 0. Mark the point (1, F(l)) on the graphs of F and F-1. Graph each of the above functions and its inverse. Mark the given point on the graph of each function.
f(x) = 2x + 3 and g(x) = 3x - 5. Find the compositions of the given functions. Which pairs of functions commute?
f(x) = 2x + 3 and h(x) = -3x - 12. Find the compositions of the given functions. Which pairs of functions commute?
F(x) = x2 + 1 and G(x) = x + l. Find the compositions of the given functions. Which pairs of functions commute?
F(x) = x2 + 1 and H(x) = -x + l. Find the compositions of the given functions. Which pairs of functions commute?
g(y) = 5y at y = 0, y = 1, and y = 4. Compute the values of the above functions at the points indicated and sketch a graph of the function.
A population of birds begins at a large value, decreases to a tiny value, and then increases again to an intermediate value. Draw graphs based on the above descriptions.
The amount of DNA in an experiment increases rapidly from a very small value and then levels out at a large value before declining rapidly to 0. Draw graphs based on the above descriptions.
Body temperature oscillates between high values during the day and low values at night. Draw graphs based on the above descriptions.
Soil is wet at dawn, quickly dries out and stays dry during the day, and then becomes gradually wetter again during the night. Draw graphs based on the above descriptions.
The number of bees b found on a plant is given by b = 2f + 1 where f is the number of flowers, ranging from 0 to about 20. Explain what might be happening when f = 0. Evaluate the above functions over the suggested range, sketch a graph of the function, and answer the biological question.
The number of cancerous cells c as a function of radiation dose r (measured in rads) is c = r - 4 for r greater than or equal to 5, and is zero for r less than 5. r ranges from 0 to 10. What is happening at r = 5 rads? Evaluate the above functions over the suggested range, sketch a graph of the
Insect development time A (in days) obeys A = 40 - T/2 where T represents temperature in °C for 10 ≤ T ≤ 40. Which temperature leads to the more rapid development? Evaluate the above functions over the suggested range, sketch a graph of the function, and answer the biological question.
Tree height h (in meters) follows the formula h = 100a/100 + a Where a represents the age of the tree in years for 0 ≤ a ≤ 1000. How tall would this tree get if it lived forever? Evaluate the above functions over the suggested range, sketch a graph of the function, and answer the biological
Graph length as a function of age.Consider the following data describing the growth of a tadpole.
h(z) = 1/5z at z = 1, z = 2, and z = 4. Compute the values of the above functions at the points indicated and sketch a graph of the function.
Graph tail length as a function of age.Consider the following data describing the growth of a tadpole.
Graph tail length as a function of length.Consider the following data describing the growth of a tadpole.
Graph mass as a function of length and then graph length as a function of mass. How do the two graphs compare?Consider the following data describing the growth of a tadpole.
The number of mosquitoes (M) that end up in a room is a function of how far the window is open (W, in cm2) according to M(W) = 5W + 2. The number of bites (B) depends on the number of mosquitoes according to B(M) = 0.5M. Find the number of bites as a function of how far the window is open. How many
The temperature of a room (T) is a function of how far the window is open (W) according to T(W) = 40 - 0.2W. How long you sleep (S, measured in hours) is a function of the temperature according to S(T) = 14 - T/5. Find how long you sleep as a function of how far the window is open. How long would
The number of viruses (V, measured in trillions or 1012) that infect a person is a function of the degree of immunosuppression (I, the fraction of the immune system that is turned off by stress) according to V(I) = 5I2. The fever (F, measured in °C) associated with an infection is a function of
The length of an insect (L, in mm) is a function of the temperature during development (T, measured in °C) according to L(T) = 10 + T/10. The volume of the insect (V, in cubic mm) is a function of the length according to V(L) = 2L3. The mass (M in milligrams) depends on volume according to M(V) =
A population of bacteria consists of two types a and b. The first follows a(t) = 1 + t2, and the second follows b(t) = 1 - 2t + t2 where populations are measured in millions and time is measured in hours. The total population is P(t) = a(t) + b(t). Each of the above measurements is the sum of two
The above-ground volume (stem and leaves) of a plant is Va(t) = 3.0t + 20.0 + t2/2, and the below-ground volume (roots) is Vb(t) = -1.0t + 40.0 where t is measured in days and volumes are measured in cm3. The total volume is V(t) = Va(t) + Vb(t).Each of the above measurements is the sum of two
Graph M as a function of a. Does this function have an inverse? Could we use mass to figure out the age of the plant?Consider the following data describing a plant.
F(r) = r2 + 5 at r = 0, r = l, and r = 4. Compute the values of the above functions at the points indicated and sketch a graph of the function.
Graph V as a function of a. Does this function have an inverse? Could we use volume to figure out the age of the plant?Consider the following data describing a plant.
Graph G as a function of a. Does this function have an inverse? Could we use glucose production to figure out the age of the plant?Consider the following data describing a plant.
Graph G as a function of M. Does this function have an inverse? What is strange about it? Could we use glucose production to figure out the mass of the plant?Consider the following data describing a plant.
The population of people is P(t) = 2.0 x 106 + 2.0 x 104t, and the mass per person W(t) (in kg) is W(t) = 80 - 0.5t.The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the above
The population is P(t) = 2.0 x 106 - 2.0 x 104t, and the mass per person W(t) is W(t) = 80 + 0.5t.The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the above exercise, find the
The population is P(t) = 2.0 x 106 + 1000t2, and the mass per person W(t) is W(t) = 80 - 0.5t.The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the above exercise, find the
The population is P(t) = 2.0 x 106 + 2.0 x 104t, and the mass per person W(t) is W(t) = 80 - 0.005t2.The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the above exercise, find the
(0, -1), (1, 1), (2, 1), (3, 5), (4, 7). Graph the given points and say which point does not seem to fall on the graph of a simple function.
(0, 5), (1, 10), (2, 8), (3, 6), (4, 4). Graph the given points and say which point does not seem to fall on the graph of a simple function.
(0, 2), (1, 3), (2, 6), (3, 11), (4, 10). Graph the given points and say which point does not seem to fall on the graph of a simple function.
The density of the apples in the previous problem is 0.8 g/cm3 and the density of the oranges is 0.95 g/cm3. What is the total volume if you add the apples to the oranges? Compute the answers by adding the given quantities.
The area of a square with side length 1.7 cm or of a disk with radius 1.0 cm. Figure out which of the following is larger.
The volume of a sphere with radius 100 m or of a 50 cm deep lake with an area of 3.0 square km. Figure out which of the following is larger.
Pressure (force per unit area) Find the dimensions of the above quantities.
Energy (force times distance) Find the dimensions of the above quantities.
The force of gravity between two objects is equal to Gm1m2/r2 where m1 and m2 are the masses of the two objects, and r is the distance between them. What are the dimensions of the gravitational constant G? Find the dimensions of the above quantities.
Using the graph of the function g(x), sketch a graph of the shifted or scaled function, say which kind of shift or scale it is, and compare with the original function.4g(x)
Using the graph of the function g(x), sketch a graph of the shifted or scaled function, say which kind of shift or scale it is, and compare with the original function.g(x) - l
Using the graph of the function g(x), sketch a graph of the shifted or scaled function, say which kind of shift or scale it is, and compare with the original function.g(x/3)
Using the graph of the function g(x), sketch a graph of the shifted or scaled function, say which kind of shift or scale it is, and compare with the original function.g(x + l)
A tree is a perfect cylinder with radius 0.5 m no matter what the height (the volume of a cylinder with height h and radius r is πhr2). Find the volumes of the above two cartoon trees (drawing a sketch can help) assuming that the height of the first is 23.1 m and that the height of the second is
A tree is a perfect cylinder with radius equal to 0.1 times the height. Find the volumes of the above two cartoon trees (drawing a sketch can help) assuming that the height of the first is 23.1 m and that the height of the second is 24.1 m. What is the ratio of the volume of the larger tree to that
A tree looks like the tree in Exercise 27, but with half the height in the cylindrical trunk and the other half in a spherical blob on top. Find the volumes of the above two cartoon trees (drawing a sketch can help) assuming that the height of the first is 23.1 m and that the height of the second
A tree looks like the tree in Exercise 27, but with 90% of the height in the cylindrical trunk and the remaining 10% in a spherical blob on top. Find the volumes of the above two cartoon trees (drawing a sketch can help) assuming that the height of the first is 23.1 m and that the height of the
A water bed that is 2 m long, 20 cm thick, and 1.5 m wide. The density of water is 1.0 g/cm3. Find the mass in kilograms of the above objects.
A spherical cow with diameter 1.3 m and density 1.3 g/cm3. Find the mass in kilograms of the above objects.
(Based on Section 1.2, Exercise 45) The number of bees b on a plant is given by b = 2f + 1 where f is the number of flowers. Suppose each flower has 4 petals. Graph the number of bees as a function of the number of petals. Change the units in the above functions, and compare a graph in the new
The number of cancerous cells c as a function of radiation dose r is c = r - 4 for r (measured in rads) greater than or equal to 5, and is zero for r less than 5 (as in Section 1.2, Exercise 46). Suppose that radiation is instead measured in millirads (1 rad = 1000 millirads). Change the units in
Insect development time A (in days) obeys A = 40 - T/2 where T represents temperature in °C for °C between 10 and 40 (as in Section 1.2, Exercise 47). Suppose that development time is measured in hours. Change the units in the above functions, and compare a graph in the new units with the
Tree height h (in meters) follows the formula h = 100a / 100 + a where a represents the age of the tree in years (as in Section 1.2, Exercise 48). Suppose that tree age is measured instead in decades. Change the units in the above functions, and compare a graph in the new units with the original
The speed of light in cm per ns (10-9 seconds or one nanosecond) (the speed of light is about 186,000 miles/second). A fast computer takes about 0.3 ns per operation. How far does light travel in the time required by one operation? Estimate the above.
Find 65 miles per hour in centimeters per second. Convert the above into the new units.
Estimate the speed that your hair grows in miles per hour. Estimate the above.
The weight of the earth in kilograms. The earth is approximately a sphere with radius 6500 km and density 5 times that of water. Estimate the above.
Suppose a person eats 2000 Kcal per day. Using the facts that 1 Kcal is approximately 4.2 Kj (a kilojoule is a unit of energy equal to 1000 joules) and 1 watt is one joule per second (a unit of power), about how many watts does a person use? Estimate the above.
The volume of all the people on earth in cubic kilometers. If a large mine is about 3 km across and 1 km deep, would they all fit? Estimate the above.
Using the fact that the density of a cell is approximately the density of water and that water weighs 1 g/cm3, estimate the number of cells in your body. The above problems give several ways to estimate the number of cells in your body. A cell is roughly a sphere 10 μm in radius, where 1 μm is
Estimate your volume in cubic meters by pretending you are shaped like a board. Pretending that cells are cubes 20 μm on a side, what do you estimate the number of cells to be by this method? The above problem gives several ways to estimate the number of cells in your body. A cell is roughly a
The brain weighs about 1.3 kg and is estimated to have about 100 billion neurons and 10 to 50 times as many other cells (glial cells). Is this consistent with our previous estimates? The above problems give several ways to estimate the number of cells in your body. A cell is roughly a sphere 10 μm
How long a piece of string would be required to go around the equator? If the string were made l.0 meters longer and stretched out all the way around, how high would it be above the surface? Does the result surprise you? The above problem regard tying string or gift-wrapping our planet, thought of
Find 2.3 grams per cubic centimeter in pounds per cubic foot. Convert the above into the new units.
How large a piece of shrink-wrap would be required to cover the entire planet? If the wrap were increased in area by l.0m2 and stretched out all around, how high would it be above the surface? Why do you think the result is so different from the previous problem? (Working this out takes a lot of
Find 9.807 m/sec2 (the acceleration of gravity) in miles per hour per second. Convert the above into the new units.
y = 2x + 3, using points with x = 1 and x = 3. For the above line, find the slopes between the two given points by finding the change in output divided by the change in input. What is the ratio of the output to the input at each of the points? Which are proportional relations? Which are increasing
A line passing through the point (-1, 6) with slope 4. Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
A line passing through the points (1, 6) and (4, 3). Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
A line passing through the points (6, 1) and (3, 4). Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
h(z) = 1/5z at z = 1, z = 2, and z = 4, as in Exercise 5. Find the slope between z = 1 and z = 2, and the slope between z = 2 and z = 4. Check that the above curves do not have constant slope by computing the slopes between the points indicated. Compare with the graphs in Exercises 5 and 6 in
F(r) = r2 + 5 at r = 0, r = 1, and r = 4, as in Exercise 6. Find the slope between r = 0 and r = 1, and the slope between r = 1 and r = 4. Check that the above curves do not have constant slope by computing the slopes between the points indicated. Compare with the graphs in Exercises 5 and 6 in
z = -5w, using points with w = 1 and w = 3. For the above line, find the slopes between the two given points by finding the change in output divided by the change in input. What is the ratio of the output to the input at each of the points? Which are proportional relations? Which are increasing and
Solve 2x + b = mx + 7 for x. Are there any values of b or m for which this has no solution? Solve the above equation for the given variable, treating the other letters as constant parameters.
Solve mx + b = 3x + 7 for x. Are there any values of b or m for which this has no solution? Solve the above equation for the given variable, treating the other letters as constant parameters.
Graph the length of a fish in inches on the horizontal axis and centimeters on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in. Most unit conversions are proportional relations. Find the slope and graph the relations between the above units.
z = 5(w - 2) + 8, using points with w = 1 and w = 3. For the above line, find the slopes between the two given points by finding the change in output divided by the change in input. What is the ratio of the output to the input at each of the points? Which are proportional relations? Which are
Graph the length of a fish in centimeters on the horizontal axis and inches on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in. Most unit conversions are proportional relations. Find the slope and graph the relations between the above units.
Graph the mass of a fish in grams on the horizontal axis and its weight in pounds on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb. Most unit conversions are proportional relations. Find the slope and graph the relations between the above units.
Graph the weight of a fish in pounds on the horizontal axis and its mass in grams on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb. Most unit conversions are proportional relations. Find the slope and graph the relations between the above units.
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