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- Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?FInd c so that the curve y = x3+ cx2+67 will have a relative extrema at (2,5). Find other extreama and detrmine if it is maximum or minimum.1) The function, y =sin(5x-3t), represents a traveling wave with height y, position x andtime t.b) When t = 0, where is maximum x in the interval [0,2π]?c) When t = 1, where is the maximum in the interval [0,2π]?d) What is the speed and direction of this wave?
- 3. Find the linear approximation of the function g(x)=3√1 +x at a =0 and use it to approximatethe numbers 3√0.95 and 3√1.1. Illustrate by graphing g and the tangent line.15) Annual U.S. imports from a certain country in the years 1996 through 2005 could be approximated by I(t) = t2 + 3.5t + 48 (1 ≤ t ≤ 9) billion dollars, where t represents time in years since 1995. Annual U.S. exports to the country in the same years could be approximated by E(t) = 0.5t2 − 1.4t + 13 (0 ≤ t ≤ 10) billion dollars. Assuming that the trends shown in the above models continue indefinitely, calculate the limits lim t→+∞ I(t) and lim t→+∞ I(t)/E(t) algebraically. (If an answer does not exist, enter DNE.) lim t→+∞ I(t) = lim t→+∞ I(t) E(t) = Interpret your answers. In the long term, U.S. imports from the other country will (select) (be rounded or rise without bound) and be times U.S. exports to the other country. Could the given models be extrapolated far into the future? Yes or NoUse the limit definition to calculate the derivative at z = a of the linear function k(z) = 5z + 16.
- The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z = ƒ(x, y) looks like. Describe your reasoning in each case. a. ƒ(x, y) = x2y2 b. ƒ(x, y) = xy2 c. ƒ(x, y) = x3y3 d. ƒ(x, y) = 1 - x2y2 e. ƒ(x, y) = x3y2 f. ƒ(x, y) =x4y4The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each ofthe following functions, so the Second Derivative Test fails there.Determine whether the function has a maximum, a minimum, orneither at the origin by imagining what the surface z = ƒ(x, y)looks like. Describe your reasoning in each case. ƒ(x, y) = x2y2Find the length of the segment of the curve y = f (x) = e ^ x between the abscissa points x = 0 and r = In 2.