Challenge Problem A Geometric Series In calculus you will learn that certain functions can be approximated by polynomial functions. We will explore one such function now.
Using graphing utility, create a table of values with
Using graphing utility, create a table of values with
Using graphing utility, create a table of values with
What do you notice about the values of the functions as more terms are added to the polynomial? Are there some values of
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Precalculus
- Water Flea F. E. Smith has studied population growth for the water flea. Let N denote the population size. In one experiment, Smith found that G, the rate of growth per day in the population, can be modeled by G=0.44N(228N)228+3.46N a. Draw a graph of G versus N. Include values of N up to 350. b. At what population level does the greatest rate of growth occur? c. There are two values of N where G is zero. Find these values of N and explain what is occurring at these population levels. d. What is the rate of population growth if the population size is 300? Explain what is happening to the population at this level.arrow_forwardAreas of Curved Regions The Monte Carlo method can be used to estimate the area under the graph of a function. The figure below shows the region under the graph of f(x)=x2, above the x-axis, between x=0 and x=1. If we choose a point in the square at random, the probability that it lies under the graph of f(x)=x2 is the area under the graph divided by the area of the square. So if we randomly select a large number of point in the square, we have numberofhitsunderthegraphnumberofhitsinthesquareareaundergraphareaofsquare Modify the program from Problem 5 to carry out this Monte Carlo simulation and approximate the required area.arrow_forwardPresent Value If you invest P dollars the present value of your investment in a fund that pays an interest rate of r, as a decimal, compounded yearly, then after t years, your investment will have a value of F dollars, which is known as the future value. The discount rare D for such an investment is given by D=1(1+r)t where t is the life, in years, of the investment. The present value of an investment is the product of the future value and the discount rate. Find a formula that gives the present value in terms of the future value, the interest rate, and the life of the investment.arrow_forward
- pH of Wine If the pH of a wine is too high, say, 4.0 or above, the wine becomes unstable and has a flat taste. a A certain California red wine has a pH of 3.2, and a certain Italian white wine has a pH of 2.9. Find the corresponding hydrogen ion concentrations of the two wines. b Which wine has the lower hydrogen ion concentration?arrow_forwardVolume of a Box A rectangular box with a volume of 22ft3 has a square base as shown below. The diagonal of the box (between a pair of opposite corners) is 1ft longerthan each side of the base. (a) If the base has sides of length x feet, shove that x62x5x4+8=0 . (b) Show that two different boxes satisfy the given conditions. Find the dimensions in each case, rounded to the nearest hundredth of a foot.arrow_forwardAreas of Curved Regions the Monte Carlo method can be used to estimate the area under the graph of a function. The figure below shows the region under the graph of f(x)=x2 above the x-axis, between x=0 and x=1. if we choose a point in the square at random, probability that it lies under the graph of f(x)=x2 is the area under the graph divided by the area of the square. So if randomly select a large of points in the square, we have Modify the program from Problem 5 to carry out this Monte Carlo simulation and mate the required area.arrow_forward
- Radius of a Shock Wave An explosion produces a spherical shock wave whose radius R expands rapidly. The rate of expansion depends on the energy E of the explosion and the elapsed time t since the explosion. For many explosions, the relation is approximated closely by R=4.16E0.2t0.4. Here R is the radius in centimeters, E is the energy in ergs, and t is the elapsed time in seconds. The relation is valid only for very brief periods of time, perhaps a second or so in duration. a. An explosion of 50 pounds of TNT produces an energy of about 1015 ergs. See Figure 2.71. How long is required for the shock wave to reach a point 40 meters 4000 centimeters away? b. A nuclear explosion releases much more energy than conventional explosions. A small nuclear device of yield 1 kiloton releases approximately 91020 ergs. How long would it take for the shock wave from such an explosion to reach a point 40 meters away? c. The shock wave from a certain explosion reaches a point 50 meters away in 1.2 seconds. How much energy was released by the explosion? The values of E in parts a and b may help you set an appropriate window. Note: In 1947, the government released film of the first nuclear explosion in 1945, but the yield of the explosion remained classified. Sir Geoffrey Taylor used the film to determine the rate of expansion of the shock wave and so was able to publish a scientific paper concluding correctly that the yield was in the 20-kiloton range.arrow_forwardLimiting values Find the limiting value of 7+a0.6t.arrow_forward
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