A bond is guaranteed to pay 100 + 10t dollars per year for 10 years, where t is in years

Use Figure 6.39 and the fact that P = 2 when t = 0 to find values of P when t = 1, 2, 3, 4 and 5. 1 2 3 4 5 1 1 t dP/dt Figure 6.39

In Problems 23, sketch two functions F such that F = f. In one case let F(0) = 0 and in the other, let F(0) = 1.

In Problems 23, sketch two functions F such that F = f. In one case let F(0) = 0 and in the other, let F(0) = 1.

Figure 6.40 shows the derivative F(x). If F(0) = 5, find the value of F(1), F(3), and F(4). 1 2 3 4 F(x) Area = 8 Area = 6 Area = 2 x Figure 6.40

Problems 56 show the derivative f of f. (a) Where is f increasing and where is f decreasing? What are the x-coordinates of the local maxima and minima of f? (b) Sketch a possible graph for f. (You dont need a scale on the vertical axis.)

Problems 56 show the derivative f of f. (a) Where is f increasing and where is f decreasing? What are the x-coordinates of the local maxima and minima of f? (b) Sketch a possible graph for f. (You dont need a scale on the vertical axis.)

Problems 78 give a graph of f(x). Graph f(x). Mark the points x1, . . . , x4 on your graph and label local maxima, local minima and points of inflection.

Problems 78 give a graph of f(x). Graph f(x). Mark the points x1, . . . , x4 on your graph and label local maxima, local minima and points of inflection.

In Problems 918, find an antiderivative. h(t) = 3t2 + 7t + 1

In Problems 918, find an antiderivative. f(t) = 2t2 + 3t3 +4t4

In Problems 918, find an antiderivative. f(x) = 6x2 8x +3

In Problems 918, find an antiderivative. q(y) = y4 + 1 y

In Problems 918, find an antiderivative. f(x) = 3x2 + 5

In Problems 918, find an antiderivative. h(z) = 1 z

In Problems 918, find an antiderivative. p(y) = 1 y + y + 1

In Problems 918, find an antiderivative. g(t) = sint

In Problems 918, find an antiderivative. f(x) = (2x + 1)3

In Problems 918, find an antiderivative. g(x) = (x+ 1)3

In Problems 1929, find the indefinite integrals. < (4t + 7) dt

In Problems 1929, find the indefinite integrals. < 3xdx

In Problems 1929, find the indefinite integrals. < (x3 x) dx

In Problems 1929, find the indefinite integrals. < (8t + 3) dt

In Problems 1929, find the indefinite integrals. < $ 5 t2 + 6 t3 %dt

In Problems 1929, find the indefinite integrals. < $8x3 + 1 x% dx

In Problems 1929, find the indefinite integrals. < (4x + 2ex)dx

In Problems 1929, find the indefinite integrals. < (5 cos x3 sinx)dx

In Problems 1929, find the indefinite integrals. < 6x2 dx

In Problems 1929, find the indefinite integrals. < (x +1)2 dx

In Problems 1929, find the indefinite integrals. < "x + 1 x # dx

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 2 1 (2x + 3) dx

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 2 1 1 t2 dt

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 1 3 2 r3 dr

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 1 0 sin d

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 2 0 "x3 3 + 2x# dx

Find the definite integrals in Problems 3035 using the Fundamental Theorem. < 2 1 1 2t dt

Use the Fundamental Theorem to find the area under f(x) = x2 between x = 1 and x = 4.

Write the definite integral for the area under the graph of f(x) = 6x2 + 1 between x = 0 and x = 2. Use the Fundamental Theorem of Calculus to evaluate it.

Use the Fundamental Theorem to find the average value of f(x) = x2 + 1 on the interval x = 0 to x = 10. Illustrate your answer on a graph of f(x).

Use the Fundamental Theorem to determine the value of b if the area under the graph of f(x) = 8x between x = 1 and x = b is equal to 192. Assume b > 1.

Find the exact area below the curve y = x3(1 x) and above the x-axis.

Find the exact area enclosed by the curve y = x2(1x)2 and the x-axis.

(a) Sketch the area represented by the improper integral = 0 xex dx. (b) Calculate =b 0 xex dx for b = 5, 10, 20. (c) The improper integral in part (a) converges. Use your answers to part (b) to estimate its value.

Consider the improper integral < 1 1 x dx. (a) Use a calculator or computer to find =b 1 1/(x) dx for b = 100, 1000, 10,000. What do you notice? (b) Find =b 1 1/(x) dx using the Fundamental Theorem of Calculus. Your answer will contain b. (c) Now take the limit as b . What does this tell you about the improper integral?

(a) Find =b 0 xex/10 dx for b = 10, 50, 100, 200. (b) Assuming that it converges, estimate the value of = 0 xex/10 dx.

A car moves with velocity, v, at time t in hours given by v(t) = 60 50t miles/hour. (a) Does the car ever stop? (b) Write an integral representing the total distance traveled for t 0. (c) Do you think the car goes a finite distance for t 0? If so, estimate that distance.

With t in years since 2000, the population, P, of the world in billions can be modeled by P = 6.1e0.012t . (a) What does this model predict for the world population in 2010? In 2020? (b) Use the Fundamental Theorem to predict the average population of the world between 2000 and 2010.

For a product, the supply curve is p = 5+0.02q and the demand curve is p = 30e0.003q, where p is the price and q is the quantity sold at that price. Find: (a) The equilibrium price and quantity (b) The consumer and producer surplus

Show graphically that the maximum total gains from trade occurs at the equilibrium price. Do this by showing that if outside forces keep the price artificially high or low, the total gains from trade (consumer surplus + producer surplus) are lower than at the equilibrium price.

InMay 1991, Car and Driver described a Jaguar that sold for $980,000. At that price only 50 have been sold. It is estimated that 350 could have been sold if the price had been $560,000. Assuming that the demand curve is a straight line, and that $560,000 and 350 are the equilibrium price and quantity, find the consumer surplus at the equilibrium price.

The demand curve for a product has equation p = 20e0.002q and the supply curve has equation p = 0.02q + 1 for 0 q 1000, where q is quantity and p is price in $/unit. (a) Which is higher, the price at which 300 units are supplied or the price at which 300 units are demanded? Find both prices. (b) Sketch the supply and demand curves. Find the equilibrium price and quantity. (c) Using the equilibrium price and quantity, calculate and interpret the consumer and producer surplus.

Calculate the present value of a continuous revenue stream of $1000 per year for 5 years at an interest rate of 9% per year compounded continuously.

At what constant, continuous rate must money be deposited into an account if the account is to contain $20,000 in 5 years? The account earns 6% interest compounded continuously.

A bond is guaranteed to pay 100 + 10t dollars per year for 10 years, where t is in years from the present. Find the present value of this income stream, given an interest rate of 5%, compounded continuously.

Determine the constant income stream that needs to be invested over a period of 10 years at an interest rate of 5% per year compounded continuously to provide a present value of $5000.

In 1980, before the unification of Germany in 1990 and the introduction of the Euro, West Germany made a loan of 20 billion Deutsche Marks to the Soviet Union, to be used for the construction of a natural gas pipeline connecting Siberia to Western Russia, and continuing toWest Germany (UrengoiUschgorodBerlin). Assume that the deal was as follows: In 1985, upon completion of the pipeline, the Soviet Union would deliver natural gas to West Germany, at a constant rate, for all future times. Assuming a constant price of natural gas of 0.10 Deutsche Mark per cubic meter, and assuming West Germany expects 10% annual interest on its investment (compounded continuously), at what rate does the Soviet Union have to deliver the gas, in billions of cubic meters per year? Keep in mind that delivery of gas could not begin until the pipeline was completed. Thus, West Germany received no return on its investment until after five years had passed. (Note: A more complex deal of this type was actually made between the two countries.)

In Problems 5658, find an antiderivative F(x) with F(x) = f(x) and F(0) = 0. Is there only one possible solution? f(x) = 2x

In Problems 5658, find an antiderivative F(x) with F(x) = f(x) and F(0) = 0. Is there only one possible solution? f(x) = 7x

In Problems 5658, find an antiderivative F(x) with F(x) = f(x) and F(0) = 0. Is there only one possible solution? < 2x x2 +1 dx

In Problems 5968, integrate by substitution. < t cos(t2) dt

In Problems 5968, integrate by substitution. < 3x2(x3 +1)4dx

In Problems 5968, integrate by substitution. < 2x x2 + 1 dx

In Problems 5968, integrate by substitution. < (x + 10)3dx

In Problems 5968, integrate by substitution. < x(x2 + 9)6dx

In Problems 5968, integrate by substitution. < tet2 dt

In Problems 5968, integrate by substitution. < ex dx

In Problems 5968, integrate by substitution. < x2(1 + 2x3)2 dx

In Problems 5968, integrate by substitution. < 1 4 x dx

In Problems 5968, integrate by substitution. < dy y + 5

Estimate =10 0 f(x)g(x) dx if f(x) = x2 and g has the values in the following table. x 0 2 4 6 8 10 g(x) 2.3 3.1 4.1 5.5 5.9 6.1

Derive the formula (called a reduction formula): < xnex dx = xnex n < xn1ex dx.

In Problems 7173 use integration by parts to evaluate the indefinite integral. Check your answer by differentiating. < y siny dy

In Problems 7173 use integration by parts to evaluate the indefinite integral. Check your answer by differentiating. < x cos xdx

In Problems 7173 use integration by parts to evaluate the indefinite integral. Check your answer by differentiating. < ln(x2) dx