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mathematics
calculus

Questions and Answers of Calculus

Follow the same steps to compare yearly, monthly, and daily compounding for a bank giving 20% interest. The procedure banks use to compute continuously compounded interest is similar to the process
Follow the same steps to compare yearly, monthly, and daily compounding for a bank giving 100% interest (in a time of severe inflation). Why do you think compound interest makes a bigger difference
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 =
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 =
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
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
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
Find the derivatives of the following functions. f(x) = x2 sin(x).
Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. sec(θ)
cos(2θ). Simplify the answer in terms of sin(2θ). Use the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the following and use the product and sum rules to compute the derivatives
sin(2θ). Simplify the answer in terms of cos(2θ). Use the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the following and use the product and sum rules to compute the derivatives
Take the derivative of cos(θ + ϕ) with respect to θ, thinking of ϕ as a constant. Simplify the answer in terms of sin(θ + ϕ). Use the angle addition formulas (Equations 2.10.1 and 2.10.2) to
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Find the derivatives of the following functions. g(x) = x2 cos(x).
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the
Find the derivative of sin-1(x). If you use the identity cos2(x) + sin2(x) = 1, you can write the answer without any trigonometric functions. We can use Theorem 2.13 to find the derivatives of the
Find the derivative of cos-1(x). Write the answer without any trigonometric functions. We can use Theorem 2.13 to find the derivatives of the inverse trigonometric functions.
Find the derivative of tan-1(x). If you use the identity 1 + tan2(x) = sec2(x), you can write the answer without any trigonometric functions We can use Theorem 2.13 to find the derivatives of the
Find the derivative of sec-1(x). Write the answer without any trigonometric functions. We can use Theorem 2.13 to find the derivatives of the inverse trigonometric functions.
y(t) = t2 sin(t) is a solution of dy/dt = 2t sin(t) + t2 cos(t). Show that the above are solutions of the given differential equation.
g(t) = cos(t) + 2 sin(t) - 3et is a solution of d4g/dt4 = g(t). Show that the above are solutions of the given differential equation.
Find the derivatives of the following functions. h(x) = sin(θ) cos(θ).
h(t) = cos(t) + 2 sin(t) - 3et is a solution of d40h/dt40 = h(t). Show that the above are solutions of the given differential equation.
Find the derivatives of the following functions (from Section 1.8, Exercises 35-38). Sketch a graph and check that your derivative has the correct sign when the argument is equal to 0.
Find the derivatives of the following functions (from Section 1.8, Exercises 35-38). Sketch a graph and check that your derivative has the correct sign when the argument is equal to 0. g(t) = 4.0 +
Find the derivatives of the following functions (from Section 1.8, Exercises 35-38). Sketch a graph and check that your derivative has the correct sign when the argument is equal to 0.
Find the derivatives of the following functions (from Section 1.8, Exercises 35-38). Sketch a graph and check that your derivative has the correct sign when the argument is equal to 0.
Consider a spring with k = 0.1 and m = 1.0. Find the period T that produces a solution of the spring equation. Is this spring stronger or weaker than one with k = 1.0, and does the oscillation have a
Consider a spring with k = 1.0 and m = 5.0. Find the period T that produces a solution of the spring equation. Does a heavier object oscillate more slowly than a light one? Consider the function p(t)
Find the derivative of Pt.Consider the combination of the temperature cycles (Subsection 1.8.3)given by
Sketch a graph of the derivative over one month. If you measured only the derivative, which oscillation would you see?Consider the combination of the temperature cycles (Subsection 1.8.3)given by
Explain each term in this equation and show that x(t) = e–t cos(t) is a solution.The spring we studied had no friction. Friction acts as a force much like the spring itself, but is proportional to
Find the derivatives of the following functions.
Graph the solution and explain what is going on. Is friction strong in this system?The spring we studied had no friction. Friction acts as a force much like the spring itself, but is proportional to
Find the derivatives of the following functions. F(z) = 3 + cos(2z - 1).
Find the derivatives of the following functions.
Find the derivatives of the following functions. f(x) = ecos(x).
Find the derivatives of the following functions. f(x) = cos(ex).
Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. tan(θ)
Using a computer or calculator, estimate the following limits. Sketch the function.
The given functions all have limits of 0 as x → 0+. For each function, find how close the input must be to 0 for the output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch a graph
The given functions all have limits of 0 as x → 0+. For each function, find how close the input must be to 0 for the output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch a graph
The given functions all have limits of 0 as x → 0+. For each function, find how close the input must be to 0 for the output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch a graph
The given functions all have limits of 0 as x → 0+. For each function, find how close the input must be to 0 for the output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch a graph
The given functions all have limits of ∞ as x → 0+. For each function, find how close the input must be to 0 for the output to be a) greater than 10, and b) greater than 100. Sketch a graph of
The given functions all have limits of ∞ as x → 0+. For each function, find how close the input must be to 0 for the output to be a) greater than 10, and b) greater than 100. Sketch a graph of
The given functions all have limits of ∞ as x → 0+. For each function, find how close the input must be to 0 for the output to be a) greater than 10, and b) greater than 100. Sketch a graph of
Using a computer or calculator, estimate the following limits. Sketch the function.
The given functions all have limits of ∞ as x → 0+. For each function, find how close the input must be to 0 for the output to be a) greater than 10, and b) greater than 100. Sketch a graph of
f(x) = 5x + 7 near x = 0. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your graph.
f(x) = 5x +7 near x = l. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your graph.
f(x) = 5x2 near x = 0. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your graph.
f(x) = 5x2 near x = 1. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your graph.
f(x) = 5x2 + 7x + 3 near x = l. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your
Using a computer or calculator, estimate the following limits. Sketch the function.
f(x) = 5x2 + 7x + 3 near x = 2. Find the average rate of change of the above function as a function of Δx, and find the limit as Δx → 0. Graph the function and indicate the rate of change on your
The volume V(T) (in cm3) follows V(T) = l + T2.Suppose we are interested in measuring the properties of a substance at temperature of absolute zero (which is 0 degrees Kelvin). However, we cannot
The hardness H(T) follows H(T) = 10.0/1 + T. Suppose we are interested in measuring the properties of a substance at temperature of absolute zero (which is 0 degrees Kelvin). However, we cannot
For each of the following populations, the instantaneous rate of change of the population size at t = 0 is exactly 1.0 million bacteria per hour. If you computed the average rate of change between t
For each of the following populations, the instantaneous rate of change of the population size at t = 0 is exactly 1.0 million bacteria per hour. If you computed the average rate of change between t
For each of the following populations, the instantaneous rate of change of the population size at t = 0 is exactly 1.0 million bacteria per hour. If you computed the average rate of change between t
For each of the following populations, the instantaneous rate of change of the population size at t = 0 is exactly 1.0 million bacteria per hour. If you computed the average rate of change between t
A piano tuner is trying to get the note A on a piano to have a frequency of exactly 440 hertz (H), or cycles per second. An electronic tuner capable of detecting a difference of x cycles per second
The army is developing satellite-based targeting systems. A system that can send a missile within y meters of its target costs 1/y2 million dollars. a. How much would it cost to hit within 10 m? b.
Suppose a body has temperature B and is cooling toward room temperature of 20°C according to the function B(t) = 20 + 17e-t where t is measured in hours. A $ 10 thermocouple can detect a difference
Using a computer or calculator, estimate the following limits. Sketch the function.(the function is defined only for positive values of x).
Some dangerously radioactive and toxic radium was dumped in the desert in 1950. It has a half-life of 50 years, and the initial level of radioactivity was r = 10.0 rads. Nobody remembers where it
We are interested in measuring the pressure at different depths below the surface of the ocean. Pressure increases by approximately l atmosphere for every 10 m of depth below the surface (for
Solar scientists want to measure the temperature inside the sun by sending in probes. Imagine that temperature increases by 1 million°C for every 10,000 km below the surface. A probe that can handle
Using a computer or calculator, estimate the following limits. Sketch the function.
Using a computer or calculator, estimate the following limits. Sketch the function.
Using a computer or calculator, estimate the following limits. Sketch the function.
Using a computer or calculator, estimate the following limits. Sketch the function.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. q(z) = (1 + z2)-2.
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
Using the given functions, find the limits by plugging in (if possible). Say whether the limit is infinity or negative infinity. Compute the value of the function 0.1 and 0.01 above and below the
How close must the input be to x = 0 for f(x) = x + 2 to be within 0.1 of 2?For the above function, find the input tolerance necessary to achieve the given output tolerance.
How close must the input be to x = 1 for f(x) = 2x + 1 to be within 0.1 of 3? For the above function, find the input tolerance necessary to achieve the given output tolerance.
How close must the input be to x = 1 for f(x) = x2 to be within 0.1 of 1? For the above function, find the input tolerance necessary to achieve the given output tolerance.
How close must the input be to x = 2 for f(x) = 5x2 to be within 0.1 of 20? For the above function, find the input tolerance necessary to achieve the given output tolerance.
Find the equilibria of the following discrete-time dynamical systems from their graphs and apply the Graphical Criterion for stability to find which are stable. Check by cobwebbing.
Graph the following discrete-time dynamical systems, find the equilibria algebraically, and check whether the stability derived from the Slope Criterion for stability matches that found with
Graph the following discrete-time dynamical systems, find the equilibria algebraically, and check whether the stability derived from the Slope Criterion for stability matches that found with
Graph the following discrete-time dynamical systems, find the equilibria algebraically, and check whether the stability derived from the Slope Criterion for stability matches that found with
Graph an updating function that lies above the diagonal both to the left and to the right of an equilibrium. The unusual equilibrium in the text has an updating function that lies below the diagonal
Graph an updating function that is tangent to the diagonal at an equilibrium but crosses from below to above. Show by cobwebbing that the equilibrium is unstable. What is the second derivative at the
Graph an updating function that is tangent to the diagonal at an equilibrium but crosses from above to below. Show by cobwebbing that the equilibrium is stable. What is the second derivative at the
Sketch the graph of an updating function that has a corner at an equilibrium and is stable. The unusual equilibrium in the text has an updating function that lies below the diagonal both to the left
Sketch the graph of an updating function that has a corner at an equilibrium and is unstable. The unusual equilibrium in the text has an updating function that lies below the diagonal both to the
Sketch the graph of an updating function that has a corner at an equilibrium and is neither stable nor unstable. The unusual equilibrium in the text has an updating function that lies below the

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