Consider the following. 32 y = f(x) %3D 16 16 32 (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 48. (i) Sample points are left endpoints. L6 (ii) Sample points are right endpoints. R6 (iii) Sample points are midpoints. M6
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- Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.(a) Estimate the area under the graph of f(x) = 4 + 4x2 from x = −1 to x = 2 using three rectangles and right endpoints.R3 = Then improve your estimate by using six rectangles.R6 = (B) Repeat part a using left endpoints L3= L6= (c) Repeat part a using midpoints M3= M6= (d) From your sketches in parts (a)-(c), which appears to be the best estimation?(a) Estimate the area under the graph of f(x) = 9 + 2x2 from x = −1 to x = 2 using three rectangles and right endpoints.R3 = Then improve your estimate by using six rectangles.R6 = (b) Repeat part (a) using left endpoints. L3= L6= (c) Repeat part (a) using midpoints. M3= M6=
- With U.S. Department of Energy data for selected years from 2000 and projected to 2030, sulphur dioxide emissions from electricity generation (in millions of short tons per year) can be modeled by E(x) = 0.0112x2 + 0.612x + 11.9 where x is the number of years past 2000. Use n = 10 equal subdivisions and right-hand endpoints to approximate (to the nearest unit) the area under the graph of E(x) between x = 5 and x = 15.Find the area under the graph of f(x) = 3x2, above the x-axis, and between x = 0 and x = 2.Find the area of the region between the curve f(x) =log2(x+2)(x+2) and the interval -2 ≤ x < 2
- find the total area between the curve g(x)=x^3 + 2x^2 -x-2 and the x-axis, from x=0 to x=2For the production function Q (K,L)= 1000 (KL3 )1/3 , Given the current input level K = 400 and L = 40. Use marginal analysis to estimate the approximate change in output which results form an increase in K of 9 units and a decrease in L of 7 units. Verify that Young′s Theorem holds?If L2 is the sum of the approximating rectangles for the area under the curve y=|cosx| from 8π to 10π where the sample points are left endpoints, then L2/π = ?
- The crash risk of an intoxicated driver relative to a similar driver with zero blood alcohol is R(x) = 51500x4.14 + 1.09 where x is the blood alcohol level as a percent (0.01 ≤ x ≤ 0.15). Find and compare dR and ΔR for x = 0.08 and dx = Δx = 0.006. [Note: x ≥ 0.08 defines "driving while intoxicated" and R(0.08) = 2.6 means that such a driver is 2.6 times more likely to be involved in an accident than a similar driver who is not impaired.]† dR = ΔR = The reproduction function for a whale is estimated to be f(p) = −0.0005p2 + 1.02p, where p and f(p) are in thousands. Find the population that gives the maximum sustainable yield, and the size of the yield. population: size of the yield: The crash risk of an intoxicated driver relative to a similar driver with zero blood alcohol is R(x) = 51500x4.14 + 1.09 where x is the blood alcohol level as a percent (0.01 ≤ x ≤ 0.15). Find and compare dR and ΔR for x = 0.08 and dx = Δx = 0.006. [Note: x ≥ 0.08 defines…Find the area under the curve y = x3 + 3x2 and the x-axis between x = 1, and x = 3.For the area under the curve f(x) = x(x+1)(x-7) above the x-axis, and between the vertical lines x=-1 and x=0,