Concept explainers
Extrema and Points of Inflection The graph of the function f consists of the three line segments joining the points (0, 0),
(a) Sketch the graph of f.
(b) Complete the table.
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
F(x) |
(c) Find the extrema of F on the interval [0, 8].
(d) Determine all points of inflection of F on the interval (0, 8).
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Calculus Loose Leaf Bundle W/webassign
- Area of a Region Sketch the region in the coordinate plane that satisfies both the inequalities x2+y29 and yx . What is the area of this region?arrow_forwardRadius 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_forwardMinimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. Suppose g(x)=f(x) where f(x)0 for all x. Explain why the local minima and maxima of f and g occur at the same values of x. Let gx be the distance between the point 3,0 and the point (x,x2) on the graph of the parabola y=x2. Express g as a function of x. Find the minimum value of the function g that you found in part b. Use the principle described in part a to simplify your work.arrow_forward
- Reaction Rates In a chemical reaction, the reaction rate R is a function of the concentraton of the product of the reaction. For a certain second-order reaction between two substances, we have the formula R=0.01x2x+22. Here x is measured in moles per cubic meter and R is measured in moles per cubic meter per second. a. Make a graph of R versus x. Include concentrations up to 100 moles per cubic meter. b. Use functional notation to express the reaction rate when the concentration is 15 moles per cubic meter, and then calculate hat value. c. The reaction is said to be in equilibrium when the reaction rate is 0. At what two concentratoins is the reaction in equilibrium?arrow_forwardCubic functions Consider the cubic function f(x) = a * x ^ 3 + b * x ^ 2 + cx + d . a. Show that f can have 0, 1, or 2 critical points . Give examples and graphs to support your argument . b. How many local extreme values can f have?arrow_forwardThe graph of the function f consists of the three line segments joining the points (0, 0), (2, −2), (6, 2), and (8, 3). The function F is defined by the integral F(x) (a) Sketch the graph of f. (b) Complete the table. (c) Find the extrema of F on the interval [0, 8]. (d) Determine all points of inflection of F on the interval (0, 8).arrow_forward
- Symmetry Principle Let R be the region under the graph of y = f (x) over the interval [−a, a], where f (x) ≥ 0. Assume that R is symmetric with respect to the y-axis. (a) Explain why y = f (x) is even—that is, why f (x) = f (−x). (b) Show that y = xf (x) is an odd function. (c) Use (b) to prove that My = 0. (d) Prove that the COM of R lies on the y-axis (a similar argument applies to symmetry with respect to the x-axis).arrow_forward1) Let f be the function defined by f(x) = 2^x and let g be defined by g(x) = 4rootx . The region R in the first quadrant is bounded by the graphs of f and g a) find the area of R. b) The region R is the base of a solid whose cross sections are squares perpendicular to the x-axis. Find the volume of this solid. c) The region R is revolves around the line x=4 to generate a solid. Find the volume of this solid.arrow_forwardIntegral Calculus Applications Suppose we have marginal revenue (MR) and marginal cost (MC) functions: MR (q) = 8 - 6q + 2q2MC (q) = 2 + 60q - q2 Determine:(a) Revenue function, assuming that, if 50 items are produced, revenue is 229,300/3.b) Cost function, assuming that, if 50 ítems are produced, revenue is 102,700/3c) Utility functiond) Intervals where utility function is increasing and decreasing.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning