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Questions and Answers of Calculus

There is a clever proof of the Mean Value Theorem from Rolle's Theorem. The idea is to tilt the function f so that it takes on the same values at the endpoints a and b. In particular, we apply
A pot is dropped from the top of a 500-ft building exactly 200 ft above your office. Must it have fallen right past your office window? Try to apply the Intermediate Value Theorem to the above
The population of bears in Yellowstone Park has increased from 100 to 1000. Must it have been exactly 314 at some time? What additional assumption would guarantee this? Try to apply the Intermediate
Prove that the discrete-time dynamical system xt+1 = cos(xt) has an equilibrium between 0 and π/2. The Intermediate Value Theorem can often be used to prove that complicated discrete-time dynamical
A lung follows the discrete-time dynamical system ct+1 = 0.25e-3ctct + 0.75γ where γ = 5.0. Show that there is an equilibrium between 0 and γ. The Intermediate Value Theorem can often be used to
A lung follows the discrete-time dynamical system ct+1 = 0.75α(ct)ct + 0.25γ where γ = 5.0 and the function α(ct) is positive, decreasing, and α(0) = 1. Show that there is an equilibrium between
A lung follows the discrete-time dynamical system ct+1 = f(ct). We know only that neither ct+1 nor ct can exceed 1 mole/liter. Use the Intermediate Value Theorem to show that this discrete-time
A farmer sets off on Saturday morning at 6 a.m. to bring a crop to market, arriving in town at noon. On Sunday she sets off in the opposite direction at 6 a.m. and returns home along the same route,
Suppose instead that the farmer sets off one morning at 6 a.m. to bring a crop to market and arrives in town at noon. Having received a great price for her crop, she buys a new car and drives home
Why must the rate of increase have been exactly 4.0 kg/yr at some time? An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a differentiable function of time.
Draw a graph of mass against time where the mass is increasing, is equal to 10.0 kg at 13 yr, and has a growth rate of exactly 4.0 kg/yr after 1 yr. An organism grows from 4 kg to 60 kg in 14 yr.
Draw a graph of mass against time where the organism reaches 10.0 kg at 1 yr, and has a growth rate of exactly 4.0 kg/yr at 13 yr. An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is
A car starts at 60 mph and slows down to 0 mph. The average speed is 20 mph after 1 h. Draw the positions of cars from the above descriptions of 1-h trips. What speed must the car achieve according
A car starts at 60 mph, steadily slows down to 20 mph, and then speeds up to 50 mph by the end of 1 h. The average speed over the whole time is 40 mph. Draw the positions of cars from the above
A car drives 60 miles in 1 h, and never varies speed by more than 10 mph. Draw the positions of cars from the above descriptions of 1-h trips. What speed must the car achieve according to the Mean
In a test, a car drives zero net distance in 1 h by switching from reverse to forward at some point. The test includes achieving the maximum possible reverse speed (20 mph) and the maximum possible
Sketch the associated figure and estimate the solution.The Marginal Value Theorem states that the best time t to leave a patch of cabbage is the solution t of the equationwhere τ is the travel time
Use the Intermediate Value Theorem to prove that there is a solution.The Marginal Value Theorem states that the best time t to leave a patch of cabbage is the solution t of the equationwhere τ is
Use the Extreme Value Theorem to show that each of the following functions has a positive maximum on the interval 0 ≤ x ≤ 1.1. f(x) = x(x - 1)ex.2. f(x) = ex - xex.3. f(x) = 5x(l - x)(2 - x) -
Find the following limits.1.2.3.4.5.6.7.8.
For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x
The following are possible absorption functions. What happens to each as c approaches infinity? Assume that all parameters take on positive values.1.2.3.4.
Find the derivatives of the following absorption functions (from Table 3.1 with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches
A bacterial population that obeys the discrete-time dynamical system bt+1 = rbt with the initial condition b0 has the solution bt = b0rt. For the following values of r and b0, state which populations
In the polymerase chain reaction used to amplify DNA, some sequences of DNA produced are too long and others are the right length. Denote the number of overly long pieces after t generations of the
Consider the discrete-time dynamical system for medication given by Mt + 1 = 0.5Mt + 1.0 with M0 = 3.1. Find the equilibrium.2. The solution is Mt = 2.0 + 0.5t × 3.0. Find the limit as t → ∞.3.
Sketch a picture of the type II response. Which of the absorption functions does it resemble? What is the optimal prey density for a predator? The amount of food a predator eats as a function of prey
Sketch a picture of the type III response. Which of the absorption functions does it resemble? What is the optimal prey density for a predator? The amount of food a predator eats as a function of
Write the equation for the optimal prey density (the value giving the maximum rate of prey capture) in terms of F'(p). Find the optimal prey density if F(p) = p. Make sure to separately consider
Draw a picture illustrating the optimal prey density in a case with a type II functional response. Now suppose that the number of prey that escape increases linearly with the number of prey (the prey
Draw a picture illustrating the optimal prey density in a case with a type III functional response. Now suppose that the number of prey that escape increases linearly with the number of prey (the
Find the optimal prey density if F(p) = p / 1 + p. Make sure to separately consider cases with c < 1 and c > 1. Now suppose that the number of prey that escape increases linearly with the number of
For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x
Find the leading behavior of the following functions at 0 and ˆž.1. f(x) = l + x.2. g(y) = y + y3.3. h(z) = z + ez.4. F(x) = 1 + 2x + 3ex.5. m(a) = 100a + 30a2 + 1/a.6.
For each of the following functions, find the leading behavior of the numerator, the denominator, and the whole function at both 0 and ˆž. Find the limit of the function at 0 and ˆž (and check
Use the method of leading behavior, L'Hopital's rule, and the method of matched leading behaviors to graph the following absorption functions.1.2.3.4.5.6.Î±(c) = 5c(1 + c)
Use the method of matched leading behaviors to graph the following Hill functions (Chapter 2, Equation 2.6.1) and their variants.1.2.3.4.
The following discrete-time dynamical systems describe the populations of two competing strains of bacteriaat+1 = satbt+1 = rbt.For the following values of the initial conditions a0 and b0, and the
Many of our absorption equations are of the formwhere α and k are positive parameters, and where r(0) = 0, limc → ∞ r(c) = ∞ and r'(c) > 0. In each of the following cases, identify
For each pair of functions, use the basic functions (when possible) to say which approaches its limit more quickly, and then check with L'Hopital's rule.1. x2 and e2x as x †’ ˆž.2. x2 and 1000x
Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you used for your tangent line approximation and the second point you used for the secant
Use the tangent line approximation to evaluate the following in two ways. First, find the tangent line to the whole function using the chain rule. Second, break the calculation into two pieces by
For the following functions, find the tangent line approximation of the two values and compare with the true value. Indicate which approximations are too high and which are too low. From graphs of
Find the third order Taylor polynomials for the following function. f(x) = x3 + 4x2 + 3x + 1 for x near 0.
Find the third order Taylor polynomials for the following function. g(x) = 4x4 + x3 + 4x2 + 3x + 1 for x near 0.
Find the third order Taylor polynomials for the following function. f(x) = 4x2 + 3x + 1 for x near 1.
Find the third order Taylor polynomials for the following function. g(x) = 4x4 + x3 + 4x2 + 3x + 1 for x near 1.
Find the third order Taylor polynomials for the following function. h(x) = ln(x) for x near 1.
Find the third order Taylor polynomials for the following function. h(x) = sin(x) for x near 0.
Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit?
Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit?
Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit?
Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit?
Find the Taylor polynomial for h(x) = cos(x) with the base point a = 0. Use your result to findFind the Taylor polynomial of degree n and the Taylor series for the above function. Add up the given
f(x) = ln(l - x) with the base point a = 0. Use your result to findFind the Taylor polynomial of degree n and the Taylor series for the above function. Add up the given series by assuming that the
Write the tangent line approximation for the numerators and denominators of the following functions and show that the result of applying L'Hopital's rule matches that of comparing the linear
Write the tangent line approximation for the numerators and denominators of the following functions and show that the result of applying L'Hopital's rule matches that of comparing the linear
Write the tangent line approximation for the numerators and denominators of the following functions and show that the result of applying L'Hopital's rule matches that of comparing the linear
Write the tangent line approximation for the numerators and denominators of the following functions and show that the result of applying L'Hopital's rule matches that of comparing the linear
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = 5c / 1 + c
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = c / 5 + c
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = 5c2 / 1 + c2
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = 5c / e2c
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = 5c / 1 + c2
Compare the tangent line approximation of the following absorption function with the leading behavior at c = 0. If they do not match, can you explain why? α(c) = 5c (1 + c)
Consider a declining population following the formula b(t) = l / 1 + t (measured in millions). Approximate the population at each of the following times using the tangent with base point t = 0, the
Consider a declining population following the formula b(t) = l / 1 + t (measured in millions). Approximate the population at each of the following times using the tangent with base point t = 0, the
Consider a declining population following the formula b(t) = l / 1 + t (measured in millions). Approximate the population at each of the following times using the tangent with base point t = 0, the
Consider a declining population following the formula b(t) = l / 1 + t (measured in millions). Approximate the population at each of the following times using the tangent with base point t = 0, the
Consider the following table giving mass as a function of Age, a (days) Mass, M (g) 0.5 ..................... 0.125 1.0 ..................... 1.000 1.5 ..................... 3.375 2.0
Consider the following table giving mass as a function of Age, a (days) Mass, M (g) 0.5 ..................... 0.125 1.0 ..................... 1.000 1.5 ..................... 3.375 2.0
Estimate T(1) using the values at t = 0 and t = 2. Consider the following table giving temperature as a function of time. Time, f Temperature, T (°C) 0.0 .......................... 0.172 1.0
Estimate T(3) using the values at t = 2 and t = 4. Consider the following table giving temperature as a function of time. Time, f Temperature, T (°C) 0.0 .......................... 0.172 1.0
Suppose r = 0.5. Even after they have been controlled, diseases can affect many individuals. Suppose a medication has been introduced, and each person infected with a new variety of influenza infects
Suppose r = 0.95. How does the total number of people infected compare with the previous problem? Even after they have been controlled, diseases can affect many individuals. Suppose a medication has
Use the quadratic approximation to estimate the following values. Compare the estimates with the exact answer.1. 2.0232. 3.0323. √4.014. √65. sin(0.02)6. cos(-0.02)
The point marked A.Try Newton's method graphically for two steps starting from the given points on the figure.
Many equations can also be solved by repeatedly applying a discrete-time dynamical system. Compare the following discrete-time dynamical systems with the Newton's method discrete-time dynamical
Many equations can also be solved by repeatedly applying a discrete-time dynamical system. Compare the following discrete-time dynamical systems with the Newton's method discrete-time dynamical
The logistic dynamical system xt+1 = rxt(1 - xt). Start from x0 = 0.75. Find the value of the parameter r for which the given discrete-time dynamical system will converge most rapidly to its positive
The Ricker dynamical system xt+1 = rxte-xt. Start from x0 = 0.75. Find the value of the parameter r for which the given discrete-time dynamical system will converge most rapidly to its positive
Find two initial values from which Newton's method fails to solve x(x - 1)(x + 1) = 0. Which starting points converge to a negative solution? As mentioned in the text, although Newton's method works
Find two initial values from which Newton's method fails to solve x3 - 6x2 + 9x - 1 = 0. Graphically indicate a third such value. As mentioned in the text, although Newton's method works incredibly
Use Newton's method to solve x2 = 0 (the solution is 0). Why does it approach the solution so slowly? As mentioned in the text, although Newton's method works incredibly well most of the time, it can
Use Newton's method to solve √|x| = 0 (the solution is 0). This is the square root of the absolute value of x. Why does the method fail? As mentioned in the text, although Newton's method works
The point marked B.Try Newton's method graphically for two steps starting from the given points on the figure.
The point marked C.Try Newton's method graphically for two steps starting from the given points on the figure.
The point marked D.Try Newton's method graphically for two steps starting from the given points on the figure.
Use Newton's method for three steps to solve the following equations. Find and sketch the tangent line for the first step, then find the Newton's method discrete-time dynamical system to check your
Use Newton's method for three steps to solve the following equations. Find and sketch the tangent line for the first step, then find the Newton's method discrete-time dynamical system to check your
Use Newton's method for three steps to solve the following equations. Find and sketch the tangent line for the first step, then find the Newton's method discrete-time dynamical system to check your
Use Newton's method for three steps to solve the following equations. Find and sketch the tangent line for the first step, then find the Newton's method discrete-time dynamical system to check your
Many equations can also be solved by repeatedly applying a discrete-time dynamical system. Compare the following discrete-time dynamical systems with the Newton's method discrete-time dynamical
With α = 0.5, r = 1, and γ = 5.0. Show that the rate of absorption is maximized at T = 1 with the absorption function A(T) = αT (Equation 3.9.2) for the above values of α, r, and γ.
The equilibrium rate of absorption. Solve for the above without substituting in a particular functional form for A(T).
Substitute the following forms for A(T) into the expressions found in Exercise 12 and compare with the results in the text. With A(T) = αT.
Substitute the following forms for A(T) into the expressions found in Exercise 12 and compare with the results in the text. With A(T) = α(1 - e-kT).
Substitute the following forms for A(T) into the expressions found in Exercise 12 and compare with the results in the text. With A(T) = αT2 / k + T2
Substitute the following forms for A(T) into the expressions found in Exercise 12 and compare with the results in the text. With A(T) = αT / k + T (a case not considered in the text).
Findfor the following forms of A(T). Do the results make sense?A(T) = Î±T.

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