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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Use the binomial theorem to compute the following. (x + 2)3
Use the derivative to sketch a graph of each of the following functions. f(x) = l - 2x + x2 for 0 ≤ x ≤ 2.
Use the derivative to sketch a graph of each of the following functions. g(x) = 4x - x2 for 0 ≤ x ≤ 5.
Use the derivative to sketch a graph of each of the following functions. h(x) = x3 - 3x for 0 ≤ x ≤ 2.
Use the derivative to sketch a graph of each of the following functions. F(x) = x + 1/x for 0 < x ≤ 2.
Try to guess functions that have the following as their derivatives. x14
F(r) = 1.5r4, where F represents the flow in cubic centimeters per second through a pipe of radius r. If r = 1, how much will a small increase in radius change the flow (try it with Δr = 0.1)? If r = 2, how will a small increase in radius change the flow? Find the derivatives of the above
The area of a circle as a function of radius is A(r) = πr2, with area measured in cm2 and radius measured in centimeters. Find the derivative of area with respect to radius. On a geometric diagram, illustrate the area corresponding to ΔA = A(r + Δr) - A(r). What is a geometric interpretation of
The volume of a sphere as a function of radius is V(r) = 4/3πr3. Find the derivative of volume with respect to radius. On a geometric diagram, illustrate the volume corresponding to ΔV = V(r + Δr) - V(r). What is a geometric interpretation of this derivative? Do the units make sense? Find the
The car starts from a stop, slowly speeds up, cruises for a while, and then abruptly stops. One car is towing another using a rigid 50-ft pole. Sketch the positions and speeds of the two cars as functions of time in the above circumstances.
The car starts from a stop, goes slowly in reverse for a short time, stops, goes forward slowly, and then more quickly. One car is towing another using a rigid 50-ft pole. Sketch the positions and speeds of the two cars as functions of time in the above circumstances.
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. f(x) = 3x2 + 3x + 1.
f(x) = (2x + 3)(-3x + 2). Find the derivatives of the above function using the product rule.
Find the derivatives of the following functions using the quotient rule.
Find the derivatives of the following functions using the quotient rule.
Find the derivatives of the following functions using the quotient rule.
f(x) = 2x + 3 and g(x) = -3x + 2. For the above function, use base point x0 = 1.0 and Δx = 0.1 to compute Δf and Δg. Find Δ (fg) (the change in the product) by computing f(x0 + Δx)g(x0 + Δx) - f(x0)g(x0). Check that Δ(fg) = g(x0)Δf + f(x0)Δg + Δf Δg. Try the same with Δx = 0.01 and see
f(x) = x2 + 2 and g(x) = 3x2 - 1. For the above function, use base point x0 = 1.0 and Δx = 0.1 to compute Δf and Δg. Find Δ (fg) (the change in the product) by computing f(x0 + Δx)g(x0 + Δx) - f(x0)g(x0). Check that Δ(fg) = g(x0)Δf + f(x0)Δg + Δf Δg. Try the same with Δx = 0.01 and see
Suppose p(x) = f(x)g(x). Test out the incorrect formula p'(x) = f'(x)g'(x) on the following functions. f(x) = x, g(x) = x2.
Suppose p(x) = f(x)g(x). Test out the incorrect formula p'(x) = f'(x)g'(x) on the following functions. f(x) = l, g(x) = x3.
Use the product rule to show that f(x)2 is also increasing. Suppose that f(x) is a positive increasing function defined for all x.
Use the quotient rule to show that 1/f(x) is decreasing. Suppose that f(x) is a positive increasing function defined for all x.
g(z) = (5z - 3)(z + 2). Find the derivatives of the above function using the product rule.
Assuming that the power rule is true for n, find d(xn+1)/dx using the product rule, and check that it too satisfies the power rule. For positive integer powers, it is possible to derive the power rule with mathematical induction. The idea is to show that a formula is true for n = l, and then that
The population P is P(t) = 2.0 × 106 + 2.0 × 104t and the weight per person W(t) is W(t) = 80 - 0.5t The total mass of a population is the product of the number of individuals and the mass of each individual. In each case, time is measured in years and mass is measured in kilograms.a. Find the
The population P is P(t) = 2.0 × 106 - 2.0 × 104t and the weight per person W(t) is W(t) = 80 + 0.5t.The total mass of a population is the product of the number of individuals and the mass of each individual. In each case, time is measured in years and mass is measured in kilograms.a. Find the
The population P is P(t) = 2.0 × 106 + 1000t2 and the weight per person W(t) is W(t) = 80 - 0.5t The total mass of a population is the product of the number of individuals and the mass of each individual. In each case, time is measured in years and mass is measured in kilograms.a. Find the total
The population P is P(t) = 2.0 × 106 + 2.0 × 104t and the weight per person W(t) is W(t) = 80 - 0.005t2.The total mass of a population is the product of the number of individuals and the mass of each individual. In each case, time is measured in years and mass is measured in kilograms.a. Find the
The above-ground volume is Va(t) = 3.0t + 20.0 and the above-ground density is pa(t) = 1.2 - 0.01?. In each of the following situations (extending Exercise 38 in Section 2.5), the mass is the product of the density and the volume. In each case, time is measured in days and density is measured in
The below-ground volume is Vb(t) = -1.0t + 40.0 and the below-ground density is pb(t) = 1.8 + 0.02t. In each of the following situations (extending Exercise 38 in Section 2.5), the mass is the product of the density and the volume. In each case, time is measured in days and density is measured in
Suppose that the fraction of chicks that survive, P(N), as a function of the number N of eggs laid, is given by the following forms (variants of the model studied in Example 2.5.14). The total number of offspring that survive is S(N) = N ∙ P(N). Find the number of surviving offspring when the
r(y) = (5y - 3)(y2 - 1). Find the derivatives of the above function using the product rule.
Suppose that the fraction of chicks that survive, P(N), as a function of the number N of eggs laid, is given by the following forms (variants of the model studied in Example 2.5.14). The total number of offspring that survive is S(N) = N ∙ P(N). Find the number of surviving offspring when the
Suppose that the fraction of chicks that survive, P(N), as a function of the number N of eggs laid, is given by the following forms (variants of the model studied in Example 2.5.14). The total number of offspring that survive is S(N) = N ∙ P(N). Find the number of surviving offspring when the
Suppose that the fraction of chicks that survive, P(N), as a function of the number N of eggs laid, is given by the following forms (variants of the model studied in Example 2.5.14). The total number of offspring that survive is S(N) = N ∙ P(N). Find the number of surviving offspring when the
M(t) = 1 + t2 and V(t) = 1 + t.Suppose that the mass M(t) of an insect (in grams) and the volume V(t) (in cm3) are known functions of time (in days).a. Find the density p(t) as a function of time.b. Find the derivative of the density.c. At what times is the density increasing?d. Sketch a graph of
M(t) = 1 + t2 and V(t) = 1 + 2t.Suppose that the mass M(t) of an insect (in grams) and the volume V(t) (in cm3) are known functions of time (in days).a. Find the density p(t) as a function of time.b. Find the derivative of the density.c. At what times is the density increasing?d. Sketch a graph of
Per capita production is 2.0 (1 - bt/1000). Sketch f(bt) for 0 ≤ bt ≤ 1000. In a discrete-time dynamical system describing the growth of a population in the absence of immigration and emigration, the final population is the product of the initial population and the per capita production.
Per capita production is 2.0 / 1 + bt/1000. Sketch f(bt) for 0 ≤ bt ≤ 2000. In a discrete-time dynamical system describing the growth of a population in the absence of immigration and emigration, the final population is the product of the initial population and the per capita production.
With n = 3 The following steps should help you to figure out what happens to the Hill function hn(x) = xn / l + xn for large values of n. a. Compute the value of the function at x = 0, x = 1, and x = 2. b. Compute the derivative and evaluate at x = 0, x = 1, and x = 2. c. Sketch a graph. d. hn(x)
s(t) = (t2 + 2)(3t2 - 1). Find the derivatives of the above function using the product rule.
With n = 10 The following steps should help you to figure out what happens to the Hill function hn(x) = xn / l + xn for large values of n. a. Compute the value of the function at x = 0, x = 1, and x = 2. b. Compute the derivative and evaluate at x = 0, x = 1, and x = 2. c. Sketch a graph. d. hn(x)
h(x) = (x + 2)(2x + 3)(-3x + 2) (apply the product rule twice). Find the derivatives of the above function using the product rule.
F(w) = (w - 1)(2w - l)(3w - 1) (apply the product rule twice).Find the derivatives of the above function using the product rule.
Find the derivatives of the following functions using the quotient rule.
Find the derivatives of the following functions using the quotient rule.
Find the derivatives of the following functions using the quotient rule.
On the figure, labela. One critical point.b. One point with a positive derivative.c. One point with a negative derivative.d. One point with a positive second derivative.e. One point with a negative second derivative.f. One point of inflection.
Find the first and second derivatives of the following function. s(x) = l - x + x2 - x3 + x4.
Find the first and second derivatives of the following function. F(z) = z(l + z)(2 + z).
Find the first and second derivatives of the following function. R(s) = (l + s2)(2 + s).
Find the first and second derivatives of the following function. f(x) = 3 + x/2x.
Find the first and second derivatives of the following function. G(y) = 2 + y/y2.
Find the first and second derivatives of the following function and use them to sketch a graph. f(x) = x-3 for x > 0.
On the figure, labela. One critical point.b. One point with a positive derivative.c. One point with a negative derivative.d. One point with a positive second derivative.e. One point with a negative second derivative.f. One point of inflection.
Find the first and second derivatives of the following function and use them to sketch a graph. g(z) = z + 1/z for z > 0.
Find the first and second derivatives of the following function and use them to sketch a graph. h(x) = (1 - x)(2 - x)(3 - x).
Find the first and second derivatives of the following function and use them to sketch a graph. M(t) = t/1 + t for t > 0.
Find the first and second derivatives of the following function and use them to sketch a graph. f(x) = 2x3 + l for -5 ≤ x ≤ 5.
Find the first and second derivatives of the following function and use them to sketch a graph. f(x) = 1/x2 for 0 < x ≤ 2.
Find the first and second derivatives of the following function and use them to sketch a graph. f(x) = 10x2 - 50x for -5 ≤ x ≤ 5.
Find the first and second derivatives of the following function and use them to sketch a graph. f(x) = x - x2 for 0 ≤ x ≤ 1.
Find the 10th derivative of x9. Some higher derivatives can be found without a lot of calculation.
Describe the graph of the 5th derivative of x5. Some higher derivatives can be found without a lot of calculation.
Is the eighth derivative of p(x) = 7x8 - 8x7 - 5x6 + 6x5 - 4x3 positive or negative? Some higher derivatives can be found without a lot of calculation.
On the figure, labela. One critical point.b. One point with a positive derivative.c. One point with a negative derivative.d. One point with a positive second derivative.e. One point with a negative second derivative.f. One point of inflection.
Find the fifth derivative of x(l + x)(2 + x)(3 + x)(4 + x). Some higher derivatives can be found without a lot of calculation.
An object on Saturn that follows p(t) = -5.2t2 - 2.0t + 50.0.The following equations give the positions as functions of time of objects tossed from towers in various exotic solar system locations. For each,a. Find the velocity and the acceleration of this object.b. Sketch a graph of the position
An object on the Sun that follows p(t) = -137t2 + 20.0t + 500.0.The following equations give the positions as functions of time of objects tossed from towers in various exotic solar system locations. For each,a. Find the velocity and the acceleration of this object.b. Sketch a graph of the position
An object on Pluto that follows p(t) = -0.325t2 - 20.0t + 500.0.The following equations give the positions as functions of time of objects tossed from towers in various exotic solar system locations. For each,a. Find the velocity and the acceleration of this object.b. Sketch a graph of the position
An object on Mercury that follows p(t) = -1.85t2 + 20.0t.The following equations give the positions as functions of time of objects tossed from towers in various exotic solar system locations. For each,a. Find the velocity and the acceleration of this object.b. Sketch a graph of the position for 0
The following graph shows the position of a roller coaster as a function of time. When is the roller coaster going most quickly? When is it accelerating most quickly? When is it decelerating most quickly?
On the figure, labela. One critical point.b. One point with a positive derivative.c. One point with a negative derivative.d. One point with a positive second derivative.e. One point with a negative second derivative.f. One point of inflection.
The following graph shows the position of a roller coaster as a function of time. When is the roller coaster going most quickly? When is it accelerating most quickly? When is it decelerating most quickly?
Per capita production is 2.0 (1 - bt / 1000). Consider values of bt less than 1000. In a model of a growing population, we find the new population by multiplying the old population by the per capita production. For each case, find the second derivative of the new population as a function of the old
Per capita production is 2.0bt (1 - bt / 1000). Consider values of bt less than 1000. In a model of a growing population, we find the new population by multiplying the old population by the per capita production. For each case, find the second derivative of the new population as a function of the
Find the second derivative of the Hill function with n = 1 and describe the curvature of the graph.We can use the second derivative to study Hill functions
Find the second derivative of the Hill function with n = 2 and describe the curvature of the graph.We can use the second derivative to study Hill functions
A function with a positive, increasing derivative. Draw graph of functions with the above properties.
A function with a positive, decreasing derivative. Draw graph of functions with the above properties.
A function with a negative, increasing (becoming less negative) derivative. Draw graph of functions with the above properties.
A function with a negative, decreasing (becoming more negative) derivative. Draw graph of functions with the above properties.
F(x) = x2 + 4ex. Compute the first and second derivatives of the above function.
h(x) = 2x2 - 2 1n(x). Compute the first and second derivatives of the above function.
g(z) = (z + 4) ln(z). Compute the first and second derivatives of the above function.
f(x) = x2 ln(x). Compute the first and second derivatives of the above function.
F(w) = ew ln(w). Compute the first and second derivatives of the above function.
Compute the first and second derivatives of the following function. F(y) = ln(y) / ey.
s(x) = ln(x2) (use a law of logs). Compute the first and second derivatives of the above function.
r(x) = ln(x2ex) (use a law of logs). Compute the first and second derivatives of the above function.
Compute the first and second derivatives of the following function.
Compute the first and second derivatives of the following function.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. f(x) = (1 - x)ex for -2 ≤ x ≤ 1.
V(y) = 1 + y + 2y2 + 3y3 - 4ey. Compute the first and second derivatives of the above function.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. g(x) = (2 - x)ex for -2 ≤ x ≤ 1.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. G(z) = ez/z2 for 1 ≤ z ≤ 3.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. F(z) = z3/ez for 0 ≤ z ≤ 5.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. L(x) = x/2 - ln(x) for 1 ≤ x ≤ 3.
Use the first and second derivatives to sketch graphs of the following functions on the given domains. Identify regions where the function is increasing, and where it is concave up. M(x) = (x + 2) ln(x) for l ≤ x ≤ 3.
f(x) = 5x. 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. Simplify with a law of exponents and factor. c. Estimate the limit by plugging in small values of h. d.
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