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calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
g(x) = e2x. After following the steps, use the fact that g(x) = ex ∙ ex to find the derivative with the product rule. We can return to the definition to find derivatives of other exponential functions. For each of the following a. Write down the definition of the derivative of this function. b.
f(x) = x2ex. Compute the first and second derivatives of the above function.
Let x* be the solution of the equation h(x*) = 0. Show that the critical point of a generalized first-order polynomial is x* - 1 and the point of inflection is x* - 2. Polynomials form a useful set of functions in part because the derivative of a polynomial is another polynomial. Another set of
x = 1 We can return to the definition to figure out the derivative of the natural log. We will first find the derivative at different values of x. For each value of x, a. Write down the definition of the derivative for ln(x). b. Plug in some small values of h to guess the limit. c. Check that your
x = 2 We can return to the definition to figure out the derivative of the natural log. We will first find the derivative at different values of x. For each value of x, a. Write down the definition of the derivative for ln(x). b. Plug in some small values of h to guess the limit. c. Check that your
Write down the definition of the derivative at x = 2, and use a law of logs to try to convert the limit into something that looks like the limit at x = 1, as in Exercise 31. Instead of plugging different values of x into the definition of the derivative for ln(x), as in Exercises 31 and 32, we can
Write down the definition of the derivative for general x, and use a law of logs to try to convert the limit into something that looks like the limit at x = 1, as in Exercise 31. Instead of plugging different values of x into the definition of the derivative for ln(x), as in Exercises 31 and 32, we
m(t) = l - t/2. Suppose a population of bacteria grows according to P(t) = 10et. Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero?
m(t) = 1 - t. Suppose a population of bacteria grows according to P(t) = 10et. Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero?
m(t) = l - t2. Suppose a population of bacteria grows according to P(t) = 10et. Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero?
m(t) = l - t2/4. Suppose a population of bacteria grows according to P(t) = 10et. Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero?
Find the first and second derivatives of the following functions (related to the gamma distribution) and sketch graphs for 0 ≤ x ≤ 2. G(x) = √xe-x.
h(x) = (2 - x2)ex. Compute the first and second derivatives of the above function.
Find the first and second derivatives of the following functions (related to the gamma distribution) and sketch graphs for 0 ‰¤ x ‰¤ 2.
The following are differential equations that could describe a bacterial population. For each, describe in words what the equation says and check that the given solution works. Say whether the solution is an increasing or decreasing function. db/dt = b(t) has solution b(t) = et.
The following are differential equations that could describe a bacterial population. For each, describe in words what the equation says and check that the given solution works. Say whether the solution is an increasing or decreasing function. db/dt = -b(t) has solution b(t) = e-t.
The following are differential equations that could describe a bacterial population. For each, describe in words what the equation says and check that the given solution works. Say whether the solution is an increasing or decreasing function. db/dt = e-b(t) has solution b(t) = ln(t).
Compute the first and second derivatives of the following function. g(x) = ex / x.
Compute the first and second derivatives of the following function. G(z) = ez / z2.
Compute the first and second derivatives of the following function. f(x) = (1 + x) / ex.
Compute the first and second derivatives of the following function. H(t) = 1 + t2 / et.
f(x) = x + 4 1n(x). Compute the first and second derivatives of the above function.
Compute the following derivatives using the chain rule. g(x) = (1 + 3x)2.
Compute the following derivatives using the chain rule. g(y) = ln(1 + y).
Compute the following derivatives using the chain rule. A(z) = ln(1 + ez).
Compute the following derivatives using the chain rule. G(s) = 8ex2.
Compute the following derivatives using the chain rule. s(w) = 4.2√1 + ew.
Compute the following derivatives using the chain rule. L(x) = ln(ln(x)).
Compute the following derivatives using the chain rule. q(y) = yy.
F(x) = 1 / 1 + ex with a) the quotient rule and b) the chain rule. Compute the derivative of each of the above function in the two ways given.
H(y) = 1 / 1 + y3 with a) the quotient rule and b) the chain rule. Compute the derivative of each of the above function in the two ways given.
g(x) = ln(3x) with a) a law of logs and b) the chain rule. Compute the derivative of each of the above function in the two ways given.
Compute the following derivatives using the chain rule. h(x) = (1 + 2x)3.
h(x) = ln(x3) with a) a law of logs and b) the chain rule. Compute the derivative of each of the above function in the two ways given.
F(x) = (1 + 2x)2 by a) expanding the binomial and taking the derivative of the polynomial and b) applying the chain rule. Compute the derivative of each of the above function in the two ways given.
F(x) = (1 + 2x)3 by a) expanding the binomial and taking the derivative of the polynomial and b) applying the chain rule. Compute the derivative of each of the above function in the two ways given.
F(x) = x3 with a) the power rule and b) the chain rule after writing F(x) using the exponential function. Compute the derivative of each of the above function in the two ways given.
F(x) = x-5 with a) the power rule and b) the chain rule after writing F(x) using the exponential function. Compute the derivative of each of the above function in the two ways given.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). f(x) = 3x + l.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). g(x) = -x + 3.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). h(x) = 2 + x3.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). F(x) = l - e-x.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). q(x) = x + x2 for x ≥ 0.
Compute the following derivatives using the chain rule. f1(t) = (1 + 3t)30.
Find the derivatives of the inverses of the following functions in two ways: first by finding the inverse and taking its derivative directly, and then by using the formula for the derivative of the inverse (Theorem 2.13). N(x) = ex2 for x ≥ 0.
Check that implicit differentiation gives the same formula for dy/dx as does solving for y in terms of x and then finding the derivative in the ordinary way. xy = 1.
Check that implicit differentiation gives the same formula for dy/dx as does solving for y in terms of x and then finding the derivative in the ordinary way. ey = x.
Take the derivative and then rewrite as a power function. We can use laws of exponents and the chain rule to check the power rule.
Find the derivative of f(x) with the chain rule. The equation for the top half of a circle is f(x) = √1 - x2. In addition to using implicit differentiation (Example 2.9.10), we can find the slope of the tangent with the chain rule, or find it geometrically.
Find the slope of the ray connecting the center of the circle at (0, 0) to the point (x, f(x)) on the circle. Then use the fact that the tangent to a circle is perpendicular to the ray to find the slope of the tangent. Check that it matches the result with the chain rule. The equation for the top
y2 + 4x4 = 4x2 at all points where x = -0.5, x = 0.5, x = -1, and x = 1. What happens at x = 0? Use implicit differentiation to find the slope of the tangent to the above relations at the given points. What do the shapes look like?
y2 + 4xy + 4x4 = 0 at all points where x = -0.5, x = 0.5, x = -1, and x = 1. What happens at x = 0? Use implicit differentiation to find the slope of the tangent to the above relations at the given points. What do the shapes look like?
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 derivative of B as a function of W. The above functional compositions
Compute the following derivatives using the chain rule. f2(t) = (1 + 2t2)15.
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 bug (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) =
Evaluate the derivative at t = 0 and after one and two half-lives. Find and explain their relationship. The amount of carbon-14 (l4C) left t years after the death of an organism is given by Q(t) = Q0e-0.000122t where Q0 is the amount left at the time of death. Suppose Q0 = 6.0 x 1010 l4C atoms/g.
Suppose a cheetah is dashing due north at a rate of 30 m/s toward a popular watering hole, and a misguided gazelle is running due east toward the same spot at a rate of 20 m/s. The cheetah begins from a distance of 120 m, while the gazelle begins from a distance of 80 m. Find equations for the
Suppose a cheetah is dashing due south at a rate of 30 m/s toward a popular watering hole, and a misguided gazelle is running due east toward the same spot at a rate of 20 m/s. The cheetah begins from a distance of 100 m, while the gazelle begins from a distance of 80 m. Find equations for the
Suppose that at some time M = 30.0, p = 0.3, dM/dt = 2.0, and dE/dt = 0.02. Find dp/dt. Why might this be a useful calculation? We can describe the energy use of an organism with two tissue types by looking at the fraction of each type rather than the total mass of each as in Example 2.9.12, by
Suppose that at some time M = 30.0, dM/dt = 2.0 and dE/dt = 0.01 without setting a value for p. Find dp/dt for p = 0.1, p = 0.5, and p = 0.9. Is there are value of p for which dp/dt = 0? Can you explain what is going on? We can describe the energy use of an organism with two tissue types by looking
Compute the following derivatives using the chain rule.
Compute the following derivatives using the chain rule.
Compute the following derivatives using the chain rule.
Compute the following derivatives using the chain rule. f(x) = e-3x.
Find the derivatives of the following functions. Note any points where the derivative does not exist.
Find the derivatives of the following functions. Note any points where the derivative does not exist.
Find the derivatives of the following function. g(x) = (4 + 5x2)6.
Find the derivatives of the following function. c(x) = (1 + 2/x)5.
Find the derivatives of the following function. p(t) = t2e2t.
Find the derivatives of the following function. s(x) = e-3x+1 +5 1n(3x).
Find the derivatives of the following function. g(y) = e3y3 + 2y2 + y.
f(t) = et cos(t). Find one critical point. Find the derivatives and other requested quantities for the above function.
g(x) = ln(1 + x2). Find one point where g(x) is decreasing. Find the derivatives and other requested quantities for the above function.
h(y) = 1 - y / (1 + y)3. Find all values where h is increasing. Find the derivatives and other requested quantities for the above function.
c(z) = e2z - 1 / z. What is c(0)? (You need to take the limit). Find the derivatives and other requested quantities for the above function.
Find the derivatives of the following functions. Note any points where the derivative does not exist.
For each of the following functions, find the average rate of change between the given base point t0 and times t0 + Δt for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. f(t) = 2 + 3t with base point t0 = 1.0.
For each of the following functions, find the equation of the secant line connecting the given base point to and times t0 + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. h(t) = t2 + 1 with base point t0 = 0.0 (based on Exercise 4).
For each of the following functions, find the equation of the secant line connecting the given base point to and times t0 + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. G(t) = e2t with base point to = 0.0 (based on Exercise 5).
For each of the following functions, find the equation of the secant line connecting the given base point to and times t0 + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. G(t) = e-t with base point to = 0.0 (based on Exercise 6).
For each of the following functions, find the average rate of change between the given base point t0 and times t0 + Δt for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. g(t) = 2 - 3t with base point t0 = 0.0.
A population of bacteria is described by the formula b(t) = 1.5t where the time t is measured in hours. For each equation for population size, find the following and illustrate on a graph. a. The population at times 0, 1, and 2. b. The average rate of change between times 0 and 1. c. The average
A population of bacteria described by the formula b(t) = 1.2t where the time t is measured in hours. For each equation for population size, find the following and illustrate on a graph. a. The population at times 0, 1, and 2. b. The average rate of change between times 0 and 1. c. The average rate
A population following b(t) = 1.5t. For each equation for population size, find the following. a. The average rate of change between times 0 and 1.0. b. The average rate of change between times 0 and 0.1. c. The average rate of change between times 0 and 0.01. d. The average rate of change between
A population following b(t) = 2.0t. For each equation for population size, find the following. a. The average rate of change between times 0 and 1.0. b. The average rate of change between times 0 and 0.1. c. The average rate of change between times 0 and 0.01. d. The average rate of change between
A population following h(t) = 5.0t2. For each equation for population size, find the following. a. The average rate of change between times 0 and 1.0. b. The average rate of change between times 0 and 0.1. c. The average rate of change between times 0 and 0.01. d. The average rate of change between
A bacterial population following b(t) = (1.0 + 2.0t)3. For each equation for population size, find the following. a. The average rate of change between times 0 and 1.0. b. The average rate of change between times 0 and 0.1. c. The average rate of change between times 0 and 0.01. d. The average rate
For the following bacterial populations, find the average rate of change during the first hour, and during the first and second half hours. Graph the data and the secant lines associated with the average rates of change. Which populations change faster during the first half hour? b(t) = 3.0(2.0t).
For the following bacterial populations, find the average rate of change during the first hour, and during the first and second half hours. Graph the data and the secant lines associated with the average rates of change. Which populations change faster during the first half hour? b(t) = e0.5t.
For the following bacterial populations, find the average rate of change during the first hour, and during the first and second half hours. Graph the data and the secant lines associated with the average rates of change. Which populations change faster during the first half hour? b(t) = 2.0e-0.5t.
For each of the following functions, find the average rate of change between the given base point t0 and times t0 + Δt for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. h(t) = 2t2 with base point t0 = 1.0.
For the following bacterial populations, find the average rate of change during the first hour, and during the first and second half hours. Graph the data and the secant lines associated with the average rates of change. Which populations change faster during the first half hour? b(t) = 3.0(0.5t).
Follow the steps in the text used to derive the approximate differential Equation 2.1.1 db/dt = 0.6697b(t) with the following values of Δt. This requires computing the value of the function b(t) = 2.0t at times separated by Δt, and finding the average rate of change between those times. Δt = 1.0.
Follow the steps in the text used to derive the approximate differential Equation 2.1.1 db/dt = 0.6697b(t) with the following values of Δt. This requires computing the value of the function b(t) = 2.0t at times separated by Δt, and finding the average rate of change between those times. Δt = 0.5.
Follow the steps in the text used to derive the approximate differential Equation 2.1.1 db/dt = 0.6697b(t) with the following values of Δt. This requires computing the value of the function b(t) = 2.0t at times separated by Δt, and finding the average rate of change between those times. Δt =
Follow the steps in the text used to derive the approximate differential Equation 2.1.1 db/dt = 0.6697b(t) with the following values of Δt. This requires computing the value of the function b(t) = 2.0t at times separated by Δt, and finding the average rate of change between those times. Δt =
The heightConsider the following data on a tree.For each of the measurements, a. Estimate the rate of change at each age. b. Graph the rate of change as a function of age. c. Find and graph the rate of change divided by the value as a function of age. d. Use these results to describe the growth of
The massConsider the following data on a tree.For each of the measurements, a. Estimate the rate of change at each age. b. Graph the rate of change as a function of age. c. Find and graph the rate of change divided by the value as a function of age. d. Use these results to describe the growth of
For each of the following functions, find the average rate of change between the given base point t0 and times t0 + Δt for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. h(t) = t2 + 1 with base point t0 = 0.0.
Write down a limit that expresses the amount of money you would get from a bank that compounded continuously, and try to guess the answer. The procedure banks use to compute continuously compounded interest is similar to the process we used to derive a differential equation. Suppose several banks
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