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mathematics
calculus

Questions and Answers of Calculus

Findfor the following forms of A(T). Do the results make sense?A(T) = Î±(1 - e-kT).
Findfor the following forms of A(T). Do the results make sense?A(T) = Î±T2.
With α = 0.5, r = 0.5, and γ = 5.0. Why is the optimal absorption lower than in Exercise 1? Show that the rate of absorption is maximized at T = 1 with the absorption function A(T) = αT (Equation
Findfor the following forms of A(T). Do the results make sense?In general, assuming A(0) = 0.
Do a complete analysis of the case A(T) = αT / k + T.
With r = 1.0 and γ = 5.0, but without picking a value for a. How does the optimal absorption depend on α? Show that the rate of absorption is maximized at T = 1 with the absorption function A(T) =
In general, without picking values for any of the parameters. 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
The equilibrium given in Equation 3.9.8. Check the above formula.
The equilibrium given in Equation 3.9.10. Check the above formula.
Î± = 0.5, r = 0.5, Î³ = 5.0, and k = 0.1. You should find that the best T is l / ˆš5.0 ‰ˆ 0.447.Find the value of T that maximizes the rate of absorption with the absorption
Find the value of T that maximizes the rate of absorption with the absorption function(Equation 3.9.9) for the following parameter values.Check thatin general.
For the functions shown:a. Sketch the derivative.b. Label local and global maxima.c. Label local and global minima.d. Find subsets of the domain with positive second derivative.
Find the Taylor polynomial of degree 2 approximating each of the following. g(x) = 1 + x / 1 + ex for x near 0.
Find the Taylor polynomial of degree 2 approximating each of the following. h(x) = x / 2 - ex for x near 0.
Combine the Taylor polynomials from the previous set of problems with the leading behavior of the functions for large x to sketch graphs. g(x)
Combine the Taylor polynomials from the previous set of problems with the leading behavior of the functions for large x to sketch graphs. h(x)
For the functions shown:a. Sketch the derivative.b. Label local and global maxima.c. Label local and global minima.d. Find subsets of the domain with positive second derivative.
Use the tangent line and the quadratic Taylor polynomial to find approximate values of the following. Write down the function or functions you use and the equation of the tangent line. Check with
Use the tangent line and the quadratic Taylor polynomial to find approximate values of the following. Write down the function or functions you use and the equation of the tangent line. Check with
Sketch graphs of the following functions. Find all critical points, and state whether they are minima or maxima. Find the limit of the function as x → ∞. (x2 + 2x)e-x for positive x.
ln(x)/(1 + x) for positive x. Do not solve for the maximum, just show that there must be one. Sketch graphs of the following functions. Find all critical points, and state whether they are minima or
Find the Taylor polynomial of degree 2 approximating each of the following. f(x) = 1 + x / 1 + x2 for x near 0.
Apply Euler's method to the following differential equations to estimate the solution at t = 1 starting from the given initial condition. First, use one step with Δt = 1, and then use two steps with
Apply Euler's method to the following differential equations to estimate the solution at t = 1 starting from the given initial condition. First, use one step with Δt = 1, and then use two steps with
A cell starts at a volume of 600 μm3 and loses volume at a rate of 2 μm3 per second.
A cell starts at a volume of 400 μm3 and gains volume at a rate of 3 μm3 per second.
A cell starts at a volume of 900 μm3 and loses volume at a rate of 2t μm3 per second.
A cell starts at a volume of 1000 μm3 and loses volume at a rate of 3t2 μm3 per second.
A snail starts crawling across a sidewalk, trying to reach the other side which is 50 cm away. The velocity of the snail t minutes after it starts is t cm/min.The following describe the velocities of
A cheetah is standing 1 m from the edge of the jungle. It starts sprinting across the savanna to attack a zebra that is 200 m from the edge of the jungle. After t seconds, the velocity of the cheetah
The cell in Exercise 13. Use a step size of Δt = 10 to estimate the volume at t = 40. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
The cell in Exercise 14. Use a step size of Δt = 5 to estimate the volume at t = 30. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
The cell in Exercise 15. Use a step size of Δt = l0 to estimate the volume at t = 30. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
The cell in Exercise 16. Use a step size of Δt = 2 to estimate the volume at t = 10. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
The snail in Exercise 17. Use a step size of Δt = 2 to estimate the position at t = 10. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
The cheetah in Exercise 18. Use a step size of Δt = 1 to estimate the position at t = 5. Apply Euler's method with the given value of Δt to the differential equation. Compare the approximate result
Use the hints to "guess" the solution of the following differential equations describing the rate of production of some chemical. In each case, check your solution, and graph the rate of change and
Use the hints to "guess" the solution of the following differential equations describing the rate of production of some chemical. In each case, check your solution, and graph the rate of change and
Mass (rate of change is growth rate). For each of the following measurements, give circumstances under which you could measure the following. a. The value but not the rate of change. b. The rate of
Sodium concentration (with rate of change equal to the rate at which sodium enters and leaves). For each of the following measurements, give circumstances under which you could measure the following.
Total chemical (with rate of change equal to the chemical production rate).For each of the following measurements, give circumstances under which you could measure the following.a. The value but not
Use the graph of the rate of change to sketch a graph of the function, starting from the given initial condition.Start from x(0) = 1.
Use the graph of the rate of change to sketch a graph of the function, starting from the given initial condition.Start from y(0) = -1.
Use the graph of the rate of change to sketch a graph of the function, starting from the given initial condition.Start from z(0) = 2.
Use the graph of the rate of change to sketch a graph of the function, starting from the given initial condition.Start from w(0) = 1.
An anti derivative that passes through the point (0, 500).From the graphs, sketch an anti derivative of the function that passes through the given point.
Find the indefinite integrals of the following function.y4 + 5y3
Find the indefinite integrals of the following function.5/x3
Find the indefinite integrals of the following function.3z3/7
Find the indefinite integrals of the following function.2 / 3√t + 3
Find the indefinite integrals of the following function.5z-1.2 - 1.2
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. dV/dt = 2t2 + 5 with V(1) = 19.0. Sketch the rate of
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. dV/dt = 2t2 + 5 with V(0) = 19.0. Sketch the rate of
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. df/dt = 5t3 + 5t with f(0) = -12.0. Sketch the rate
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. dg/dt = -3t + t2 with g(0) = 10.0. Sketch the rate
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. dM/dt = t2 + 1/t2 with M(3) = 10.0. Sketch the rate
An anti derivative that passes through the point (50, 5000).From the graphs, sketch an anti derivative of the function that passes through the given point.
Use indefinite integrals to solve the following differential equations. Sketch a graph of the rate of change and the solution on the given domain. dp/dt = 5t3 + 5/t2 with p(1) = 12.0. Sketch the rate
Use the functions f(x) = x2 and g(x) = x3 to show that the product of integrals is not equal to the integral of the product. There are no simple integral versions of product and quotient rules for
Use the functions f(x) = x2 and g(x) = x3 to show thatThere are no simple integral versions of product and quotient rules for derivatives. Use the given functions to show that proposed rule does not
Suppose a cell is taking water into two vacuoles. Let V1 denote the volume of the first vacuole and V2 the volume of the second. In each of the following cases,i. Solve the given differential
Suppose a cell is taking water into two vacuoles. Let V1 denote the volume of the first vacuole and V2 the volume of the second. In each of the following cases,a. Solve the given differential
Suppose organisms grow in mass according to the differential equationdM /dt = αtnwhere M is measured in grams and t is measured in days. For each of the following values for n and α, finda. The
Suppose organisms grow in mass according to the differential equationdM /dt = αtnwhere M is measured in grams and t is measured in days. For each of the following values for n and α, finda. The
On Earth where a = -9.8 m/s2Suppose an object is thrown from a height of h = 100 m with velocity v = 5.0 m/s (upward) to find its trajectory in a local gravitational field of strength a. For the
On the moon where a = -1.62 m/s2 Suppose an object is thrown from a height of h = 100 m with velocity v = 5.0 m/s (upward) to find its trajectory in a local gravitational field of strength a. For the
On Jupiter where a = -22.88 m/s2 Suppose an object is thrown from a height of h = 100 m with velocity v = 5.0 m/s (upward) to find its trajectory in a local gravitational field of strength a. For the
An anti derivative that passes through the point (100, 3000).From the graphs, sketch an anti derivative of the function that passes through the given point.
On Mars' moon Deimos where a = -2.15 × 10-3 m/s2 Suppose an object is thrown from a height of h = 100 m with velocity v = 5.0 m/s (upward) to find its trajectory in a local gravitational field of
Object 1, with initial condition p(0) = 10.0. To find the formula for the velocities, note that they increase linearly in time.The velocities of four objects are measured at discrete times.Use
Object 2, with initial condition p(0) = 10.0. To find the formula for the velocities, note that they decrease linearly in time.The velocities of four objects are measured at discrete times.Use
Object 3, with initial condition p(0) = 10.0. To find the formula for the velocities, compare them with the perfect square numbers.The velocities of four objects are measured at discrete times.Use
Object 4, with initial condition p(0) = 10.0. The velocities follow a quadratic equation of the form v(t) = t2/2 + at + b for some value of a.The velocities of four objects are measured at discrete
Object 1, with initial condition p(0) = 10.0. Consider again the velocities of four objects used in the previous set of problems. There is a more accurate variant of Euler's method that approximates
Object 2, with initial condition p(0) = 10.0. Consider again the velocities of four objects used in the previous set of problems. There is a more accurate variant of Euler's method that approximates
Object 3, with initial condition p(0) = 10.0. Consider again the velocities of four objects used in the previous set of problems. There is a more accurate variant of Euler's method that approximates
Object 4, with initial condition p(0) = 10.0. Consider again the velocities of four objects used in the previous set of problems. There is a more accurate variant of Euler's method that approximates
An anti derivative that passes through the point (0, 2500).From the graphs, sketch an anti derivative of the function that passes through the given point.
An anti derivative that passes through the point (100, 1000).From the graphs, sketch an anti derivative of the function that passes through the given point.
An anti derivative that passes through the point (0, 1000).From the graphs, sketch an anti derivative of the function that passes through the given point.
Find the indefinite integrals of the following function.7x2
Find the indefinite integrals of the following function.10t9 + 6t5
Find the indefinite integrals of the following function.72t + 5
Find the indefinite integrals of the following function. 3/z2 + z2/3
Use substitution, if needed, to find the indefinite integrals of the following functions. Check with the chain rule. (1 + 2t)-4
Use substitution, if needed, to find the indefinite integrals of the following functions. Check with the chain rule. 1 / 4 + t
Use substitution, if needed, to find the indefinite integrals of the following functions. Check with the chain rule. 1 / 1 + 4t
Use substitution to find the indefinite integrals of the following function. ex / 1 + ex
Use substitution to find the indefinite integrals of the following function. t / 1 + t
Find the indefinite integrals of the following function. 3ex + 2x3
Use integration by parts to evaluate the following. Check your answer by taking the derivative. ∫ xe2xdx
Use integration by parts to evaluate the following. Check your answer by taking the derivative. ∫ x cos(3x)dx
Use integration by parts to evaluate the following. Check your answer by taking the derivative. ∫ x2exdx
Use integration by parts to evaluate the following. Check your answer by taking the derivative. ∫ x2e-xdx
Use integration by partial fractions to compute the following indefinite integrals.
Use integration by partial fractions to compute the following indefinite integrals.
Substitute x = tan(θ) to integrate 1/1 + x2. Integration by substitution, in combination with trigonometric identities, can be used to integrate some surprising functions.
Substitute x = sin(θ) to integrate 1/√1 - x2. Integration by substitution, in combination with trigonometric identities, can be used to integrate some surprising functions.
The result of Section 2.10, Exercise 25 gives the derivative of tan-1(x). Use integration by parts and a substitution to find ∫ tan-1(x)dx. Integration by parts along with substitution can be used
Find the indefinite integrals of the following function. ex + 1/x

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