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mathematics
calculus

Questions and Answers of Calculus

The definite integral can be used to find the area between two curves. In each case, a. Sketch the graphs of the two functions over the given range, and shade the area between the curves. b. Sketch
The definite integral can be used to find the area between two curves. In each case, a. Sketch the graphs of the two functions over the given range, and shade the area between the curves. b. Sketch
The definite integral can be used to find the area between two curves. In each case, a. Sketch the graphs of the two functions over the given range, and shade the area between the curves. b. Sketch
The definite integral can be used to find the area between two curves. In each case, a. Sketch the graphs of the two functions over the given range, and shade the area between the curves. b. Sketch
Find the average value of the following functions over the given range. Sketch a graph of the function along with a horizontal line at the average to make sure that your answer makes sense. x2 for 0
Use substitution to evaluate the following definite integrals.
Find the average value of the following functions over the given range. Sketch a graph of the function along with a horizontal line at the average to make sure that your answer makes sense. 1/x for
Find the average value of the following functions over the given range. Sketch a graph of the function along with a horizontal line at the average to make sure that your answer makes sense. x - x3
Find the average value of the following functions over the given range. Sketch a graph of the function along with a horizontal line at the average to make sure that your answer makes sense. sin(2x)
Find the area under the curve g(x) = x ln(x) for 1 ≤ x ≤ 2. Sketch a graph to see if your answer makes sense. Use integration by parts to evaluate the following as definite integrals.
Find the area under the curve g(x) = x sin(27πx) for 0 ≤ x ≤ 2. Sketch a graph to see if your answer makes sense. Use integration by parts to evaluate the following as definite integrals.
We have used little vertically oriented rectangles to compute areas. There is no reason why little horizontal rectangles cannot be used. Here are the steps to find the area under the curve y = f(x)
We have used little vertically oriented rectangles to compute areas. There is no reason why little horizontal rectangles cannot be used. Here are the steps to find the area under the curve y = f(x)
Use the fact that the area of a circle of radius r is πr2 to find the volume of a cone of height 1 that has radius r at a height r. Think of the cone as being built of a stack of little circular
Break a sphere of radius r into horizontal discs to find the volume. The trick is to figure out the area of each disc at height z where z ranges from -r to r. There is often more than one way to
Show that l(6) - l(3) = l(2). (Use the summation property of the definite integral to write the difference as an integral, and then use the substitution y = x/3.)Some books define the natural log
Use substitution to evaluate the following definite integrals.
Find the integral from a to 2a by following the same steps. (Make the substitution y = x/a.)Some books define the natural log function with the definite integral as the function 1(a) for whichUsing
Some books define the natural log function with the definite integral as the function 1(a) for whichUsing this definition, we can prove the laws of logarithms.Show that l(102) = 2 ˆ™ l(10). (Try
Some books define the natural log function with the definite integral as the function 1(a) for whichUsing this definition, we can prove the laws of logarithms.Show that l(ab) = b ˆ™ l(a). (Try the
Suppose a math class has four equally weighted tests. A student gets 60 on the first test, 70 on the second, 80 on the third, and 90 on the last. The average of a step function computed with the
A math class has 20 students. In a quiz worth 10 points, 4 students get 6, 7 students get 7, 5 students get 8, 3 students get 9, and 1 student gets 10. The average of a step function computed with
Find the total amount of water entering during the first 15 hr, from t = 0 to t = 15. Find the average rate at which water entered during this time. Suppose water is entering a tank at a rate of g(t)
Find the total amount and average rate from t = 15 to t = 24. Suppose water is entering a tank at a rate of g(t) = 360t - 39t2 + t3 where g is measured in liters per hour and t is measured in hours.
Find the total amount and average rate from t = 0 to t = 24. Suppose water is entering a tank at a rate of g(t) = 360t - 39t2 + t3 where g is measured in liters per hour and t is measured in hours.
Suppose that energy is produced at a rate of E(t) = |g(t)| in J/h (Joules per hour). Find the total energy generated from t = 0 to t = 24. Find the average rate of energy production. Check that g(t)
Several very skinny 2.0 m long snakes are collected in the Amazon. Each has density of p(x) given by the following formulas, where p is measured in g/cm and x is measured in centimeters from the tip
Use substitution to evaluate the following definite integrals.
Several very skinny 2.0 m long snakes are collected in the Amazon. Each has density of p(x) given by the following formulas, where p is measured in g/cm and x is measured in centimeters from the tip
Suppose that the number of A's per thousand increases linearly from 150 at one end of the DNA strand to 300 at the other. The number of C's per thousand decreases linearly from 350 at one end to 200
Suppose that the number of A's per thousand increases linearly from 200 at one end of the DNA strand to 250 at the other. The number of C's per thousand increases linearly from 250 at one end to 300
Water is entering at a rate of t3cm3/s. Suppose water is entering a series of vessels at the given rate. In each case, find the total amount of water entering during the first second, and the average
Water is entering at a rate of √t cm3/s. Suppose water is entering a series of vessels at the given rate. In each case, find the total amount of water entering during the first second, and the
Water is entering at a rate of t cm3/s. Suppose water is entering a series of vessels at the given rate. In each case, find the total amount of water entering during the first second, and the average
Suppose water is entering a series of vessels at the given rate. In each case, find the total amount of water entering during the first second, and the average rate during that time. Compare the
Use substitution to evaluate the following definite integrals.
Use substitution to evaluate the following definite integrals.
Find the areas under the following curves. If you use substitution, draw a graph to compare the original area with that in transformed variables. Area under h(x) = ex/2 from x = 0 to x = ln(2).
Which function approaches 0 faster as x approaches infinity: e-x or 1/x? Many improper integrals can be evaluated by comparing functions with the method of leading behavior. State which of the given
Evaluate the following improper integrals or say why they don't converge.
Use the comparison test to deduce whether the following improper integrals converge. If they do, find an upper bound on the value.
Use the comparison test to deduce whether the following improper integrals converge. If they do, find an upper bound on the value.
Use the comparison test to deduce whether the following improper integrals converge. If they do, find an upper bound on the value.
Use the comparison test to deduce whether the following improper integrals converge. If they do, find an upper bound on the value.
Which function approaches 0 faster as x approaches infinity: 1/1 + x2 or 1/1 + x. Many improper integrals can be evaluated by comparing functions with the method of leading behavior. State which of
The method of leading behavior can be used to deduce whether some improper integrals converge. Choose the leading behavior of the denominator of each function and compare with the results using the
Compare the following series with the given integral to determine whether the sum approaches infinity.Compare
Compare the following series with the given integral to determine whether the sum approaches infinity.Compare
Compare the following series with the given integral to determine whether the sum approaches infinity.Compare
Compare the following series with the given integral to determine whether the sum approaches infinity.Compare
The volume of a cell is increasing at a rate of 100/(1 + t)2 μm3/s, starting from a size of 500 μm3. Write pure-time differential equations to describe the above situations, find out what happens
The concentration of a toxin in a cell is increasing at a rate of 50e-2t μmol/L/s, starting from a concentration of 10 μmol/L. If the cell is poisoned when the concentration exceeds 30 μmol/L,
A population of bacteria is increasing at a rate of 1000 / (2 + 3t)0.75 bacteria per hour, starting from a population of 106. Could this sort of growth be maintained indefinitely? When would the
Which function approaches infinity faster as x approaches 0: 1/x2 or 1/x. Many improper integrals can be evaluated by comparing functions with the method of leading behavior. State which of the given
A population of bacteria is increasing at a rate of 1000 / (2 + 3t)1.5 bacteria per hour, starting from a population of 1000. Could this sort of growth be maintained indefinitely? Would the
Which function approaches infinity faster as x approaches 0: 1/x or 1/√x. Many improper integrals can be evaluated by comparing functions with the method of leading behavior. State which of the
Evaluate the following improper integrals or say why they don't converge.
Evaluate the following improper integrals or say why they don't converge.
Evaluate the following improper integrals or say why they don't converge.
The voltage v of a neuron follows the differential equationover the course of 100 ms, where t is measured in ms and v in millivolts. We start at v(0) = -70.a. Sketch a graph of the rate of change.
Consider again the differential equation in the previous problem,with v(0) = -70.a. Use Euler's method to estimate the voltage after 1 ms, and again 1 ms after that.b. Estimate the voltage after 2 ms
A neuron in your brain sends a charge down an 80 cm long axon (a long skinny thing) toward your hand at a speed of 10 m per second. At the time when the charge reaches your elbow, the voltage in the
The phase line in Exercise 12.From the given phase-line diagram, sketch a possible graph of the rate of change of x as a function of x.
Graph the rate of change as a function of the state variable and draw the phase-line diagram for the following differential equations.Graph for -2 ‰¤ x ‰¤ 2.
Graph the rate of change as a function of the state variable and draw the phase-line diagram for the following differential equations.Graph for -2 ‰¤ x ‰¤ 2.
Graph the rate of change as a function of the state variable and draw the phase-line diagram for the following differential equations.Graph for -2 ‰¤ y ‰¤ 2.
Graph the rate of change as a function of the state variable and draw the phase-line diagram for the following differential equations.Graph for 0
Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h. The differential equation describing such a population is db/dt = λb - h.
Find the equilibria of the following autonomous differential equations.
Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h. The differential equation describing such a population is db/dt = λb - h. Discuss.
Check that your arrows are consistent with the behavior of b(t) at b = 10 and b = 1000. Find the equilibria, graph the rate of change db/dt as a function of b, and draw a phase-line diagram for the
Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 5000. Find the equilibria, graph the rate of change db/dt as a function of b, and draw a phase-line diagram for the
Check that your arrows are consistent with the behavior of b(t) at b = 100 and b = 300. Find the equilibria, graph the rate of change db/dt as a function of b, and draw a phase-line diagram for the
Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 3000. Find the equilibria, graph the rate of change db/dt as a function of b, and draw a phase-line diagram for the
The model in Section 5.1, Exercise 33. Check that the direction arrow is consistent with the behavior of C(t) at C = Г. Find the equilibria, graph the rate of change dC/dt as a function of C, and
Check that the direction arrow is consistent with the behavior of C(t) at C = Г. Find the equilibria, graph the rate of change dC/dt as a function of C, and draw a phase-line diagram for the
What happens to a solution starting from a small, but positive, value of p? Find the equilibria, graph the rate of change dp/dt as a function of p, and draw a phase-line diagram for the following
What happens to a solution starting from a small, but positive, value of p? Find the equilibria, graph the rate of change dp/dt as a function of p, and draw a phase-line diagram for the following
Suppose the population size of some species of organism follows the modelwhere N is measured in hundreds. Why might this population behave as it does at small values? This is another example of the
Find the equilibria of the following autonomous differential equations.
Suppose the population size of some species of organism follows the modelwhere N is measured in hundreds. What is the critical value below which this population is doomed to extinction (as in
The drag on a falling object is proportional to the square of its speed. In a differential equationwhere v is speed, a is acceleration, and D is drag. Suppose that a = 9.8 m/s2 and that D = 0.0032
Consider the same situation as in Exercise 31 but for a skydiver diving head down with her arms against her sides and her toes pointed, thus minimizing drag. The drag D is reduced to D = 0.00048 per
According to Torricelli's law of draining, the rate that a fluid flows out of a cylinder through a hole at the bottom is proportional to the square root of the depth of the water. Let y represent the
Write a differential equation describing the depth of water in a cylinder where water enters at a rate of 4.0 cm/s but drains out as in Exercise 33. Use your phase-line diagram to sketch solutions
One of the most important differential equations in chemistry uses the Michaelis-Menton or Monod equation. Suppose S is the concentration of a substrate that is being converted into a product.
Write a differential equation describing the amount of substrate if substrate is added at rate R but is converted into product as in Exercise 35. Find the equilibrium, and draw the phase-plane
Suppose that energy is used at a rate proportional to the mass. In this case,where V represents the volume in cubic centimeters and t is time measured in days. The first term says that surface area
Suppose that energy is used at a rate proportional to the mass to the 3/4 power (as in Example 1.7.24). In this case,Find the units of a2 if V is measured in cm3 and t is measured in days (they
Find the equilibria of the following autonomous differential equations.
Find the equilibria of the following autonomous differential equations that include parameters.
Find the equilibria of the following autonomous differential equations that include parameters.
Find the equilibria of the following autonomous differential equations that include parameters.
Find the equilibria of the following autonomous differential equations that include parameters.
From the following graphs of the rate of change as a function of the state variable, draw the phase-line diagram.
From the following graphs of the rate of change as a function of the state variable, identify stable and unstable equilibria by checking whether the rate of change is an increasing or decreasing
Find the stability of the equilibria of the following autonomous differential equations that include parameters.Suppose that Î² 1.
Consider the differential equation dx/dt = x2. Find the equilibrium, graph the rate of change as a function of x, and draw the phase-line diagram. Would you consider the equilibrium to be stable or
Consider the differential equation dx/dt = -(1 - x)4. Find the equilibrium, graph the rate of change as a function of x, and draw the phase-line diagram. Would you consider the equilibrium to be
Graph a rate-of-change function that has a slope of 0 at the equilibrium but the equilibrium is stable. What is the sign of the second derivative at the equilibrium? What is the sign of the third

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