Review the following concepts if needed:
Part A The use of a calculator is not allowed.
- Find the value of c as stated in the Mean Value Theorem for Integrals for f (x) = x^{3} on [2, 4].
- The graph of f is shown in Figure 13.9-1. Find the average value of f on [0, 8].
- The position function of a particle moving on a coordinate line is given as s (t) = t^{2} – 6t – 7, 0 ≤ t ≤ 10. Find the displacement and total distance traveled by the particle from 1 ≤ t ≤ 4.
- The velocity function of a moving particle on a coordinate line is v(t) = 2t + 1 for 0 ≤ t ≤ 8. At t =1, its position is –4. Find the position of the particle at t = 5.
- The rate of depreciation for a new piece of equipment at a factory is given as p(t) = 50t – 600 for 0 ≤ t ≤ 10, where t is measured in years. Find the total loss of value of the equipment over the first 5 years.
- If the acceleration of a moving particle on a coordinate line is a(t) = – 2 for 0 ≤ t ≤ 4, and the initial velocity v_{0} = 10, find the total distance traveled by the particle during 0 ≤ t ≤ 4.
- The graph of the velocity function of a moving particle is shown in Figure 13.9-2. What is the total distance traveled by the particle during 0 ≤ t ≤ 12?
- If oil is leaking from a tanker at the rate of f(t) = 10e^{0.2t} gallons per hour where t is measured in hours, how many gallons of oil will have leaked from the tanker after the first 3 hours?
- The change of temperature of a cup of coffee measured in degrees Fahrenheit in a certain room is represented by the function f (t)= – cos for 0 ≤ t ≤ 5, where t is measured in minutes. If the temperature of the coffee is initially 92?F, find its temperature after the first 5 minutes.
- If the half-life of a radioactive element is 4500 years, and initially there are 100 grams of this element, approximately how many grams are left after 5000 years?
- Find a solution of the differential equation: = x cos (x^{2}); y (0) = π.
- If = x – 5 and at x =0, y' = – 2 and y =1, find a solution of the differential equation.
Part B Calculators are allowed.
- Find the average value of y = tan x from x = to x =
- The acceleration function of a moving particle on a straight line is given by a(t)=3e^{2t}, where t is measured in seconds, and the initial velocity is . Find the displacement and total distance traveled by the particle in the first 3 seconds.
- The sales of an item in a company follow an exponential growth/decay model, where t is measured in months. If the sales drop from 5000 units in the first month to 4000 units in the third month, how many units should the company expect to sell during the seventh month?
- Find an equation of the curve that has a slope of at the point (x, y) and passes through the point (0, 4).
- The population in a city was approximately 750,000 in 1980, and grew at a rate of 3% per year. If the population growth followed an exponential growth model, find the city's population in the year 2002.
- Find a solution of the differential equation 4e^{y} = y' – 3xe ^{y} and y(0) = 0.
- How much money should a person invest at 6.25% interest compounded continuously so that the person will have $50,000 after 10 years?
- The velocity function of a moving particle is given as v(t) = 2 – 6e^{–t}, t = 0 and t is measured in seconds. Find the total distance traveled by the particle during the first 10 seconds.
- Draw a slope field for the differential equation = x – y.
- A rumor spreads through an office of fifty people at a model by On day zero, one person knows the rumor. Find the model for the population at time t, and use it to predict when more than half the people in the office will have heard the rumor.
- A college dormitory that houses 200 students experiences an outbreak of influenza. The illness is recognized when two students are diagnosed on the same day. The residents are quarantined to restrict the infection to this one building. On the fifth day of the outbreak, 12 students are ill. Use a logistic model to describe the course of infection and predict the number of infected students on day 10.
- Use Euler's Method with a step size of Δx = 0.1 to compute y(.5) if y(x) is the solution of the differential equation = x^{2} – y^{3} with the condition y(0) = 1.
- Use Euler's Method with a step size of Δx = 0.5 to compute y(3) if y (x) is the solution of the differential equation y – 2x with initial condition y (0) = 1.
(Calculator) indicates that calculators are permitted.
- If 3e ^{y} =x ^{2} y, find .
- Evaluate .
- The graph of a continuous function f which consists of three line segments on [–2, 4] is shown in Figure 13.10-1. If ,
- Find F (–2) and F (0).
- Find F' (0) and F' (2).
- Find the value of x such that F has a maximum on [–2, 4].
- On which interval is the graph of F concave upward?
- (Calculator) The slope of a function y = f (x) at any point (x, y) is and f(0)=2.
- Write an equation of the line tangent to the graph of f at x =0.
- Use the tangent in part (a) to find the approximate value of f(0.1).
- Find a solution y = f(x) for the differential equation.
- Using the result in part (c), find f (0.1).
- (Calculator) Let R be the region in the first quadrant bounded by f(x) = e ^{x} – 1 and g(x)=3 sin x.
- Find the area of region R.
- Find the volume of the solid obtained by revolving R about the x -axis.
- Find the volume of the solid having R as its base and semicircular cross sections perpendicular to the x -axis.
- An object traveling on a path defined by {x(θ), y(θ)} has an acceleration vector of {sin θ, –cos θ}. If the velocity of the object at time is {–1, 0} and the initial position of the object is the origin, find the position when θ = π.
- A projectile follows a path defined by x = t – 2, y = sin^{2} t on the interval 0 ≤ t ≤ π. Find the point at which the object reaches its maximum y -value.
Solutions for these practice problems can be found at: Solutions to Applications of Definite Integrals Practice Problems for AP Calculus
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