In Exercises 19–24, calculate the average rate of change of the given function f over the intervals
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Applied Calculus
- The total public debt D (in trillions of dollars) in the United States at the beginning of each year from 2000 through 2008 can be approximated by the model D = 0.032t2 + 0.21t + 5.6, 0 ≤ t ≤ 8 where t represents the year, with t = 0 corresponding to 2000. t D 0 5.6 1 5.842 2 6.148 3 6.518 4 6.952 5 7.45 6 8.012 7 8.638 8 9.328 (b) Verify your result from part (a) algebraically and graphically. (Round your answer to two decimal places.)t =arrow_forwardProve that the straight line y=3-2x intersects the exponential curve y=e^x at a point whose x-coordinate belongs to the interval (0,1).arrow_forwardFind the Linearization L(x) of the function at a. a) f(x) = x^2 -x+2, a=3.arrow_forward
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- Consider the following function. g(x)=x^3-5 Find its average rate of change over the interval [-2,2]arrow_forwardThe analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is con-tinuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. a. If t represents time in years and y represents tooth size, use the condition that y = 0.99y0 when t = 1000 to find the value of k in the equation y = y0 ekt. Then use this value of k to answer the following questions. b. In about how many years will human teeth be 90% of their present size? c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?arrow_forwardFind the linearization of the function f(x)=ln(2x+3) at x=-1arrow_forward
- a. At what time t does the cave's salamander population reach its maximum?arrow_forwardEstimate the derivative from the table of average rates of change. HINT [See discussion at the beginning of the section.] h 1 0.1 0.01 0.001 0.0001 Avg. Rate ofChange of f over[7, 7 + h] 8 4.3 4.03 4.003 4.0003 h −1 −0.1 −0.01 −0.001 −0.0001 Avg. Rate ofChange of f over[7 + h, 7] 2 3.7 3.97 3.997 3.9997 Estimate f '(7). f '(7) =arrow_forwardWhich of the following best models the population that is continuously decreasing at 2% per year starting at 25,000? A) f(t) = 25000e02t B) f(t) = 25000e-0.02t C) f(t) = 25000e98t D) f(t) = 25000(0.98)t E) f(t) = 25000(-0.02)tarrow_forward
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