Solve the 1-dimensional zero ends heat equation problem with c = 1, L = 2, and initial temperature X if 0≤x≤1 2 x if 1 x ≤ 2 ƒ (x) = { = { 2 -

in his model for storage and shipping costs for a manufacturing process, a researcher develops the following formula c=100(100+9x+144/x), where C is the most storage and transportation for up to 100 days of operation and load of material is moved every "x" days. Find the value of x for which C is a minimum. (This is a calculus optimization problem, please show all work and add the chart for the first derivative test just like this example I am giving here, do the work like the example and put that chart too)

- 2. Suppose ƒ'(x) = (x − 1)¹⁰(x + 2)9⁹99 (x+3)(x + 5)³(x − 2)¹2000. Find all values of x for which f(x) attains a local maximum or minimum. Hint: you can reason about where f'(x) is positive or negative without actually computing the full value of f'(x).

Suppose it is know that the function f(x) = has a maximum value of 2 attained at x = 6. Can you determine the unknowns a and b? a = b | = axebx2 = AY

3. If f(x) = kx³ -1 and f¹(15) = 2, find k_

From the function f(x)= x3 -6x2 +9x -3Determines monotony, maximum and minimum points, inflection and concavities

3. Show that x³ + 2x = -1 has a solution in an interval of length. You may find it helpful to complete the following table. Interval Length of Midpoint Interval 1 A functional Value Conclusion at Midpoint

Suppose the rate of oil imports to the United States from a certain country can be approximated by r(t) = −5t2 + 40t + 900 million barrels per year    (0 ≤ t ≤ 8), where t is time in years since the start of 2000. During that time, the price of oil was approximately† p(t) = 25e0.1t dollars per barrel. Obtain an expression for the total oil revenue  R(x)  the country earned from the United States since the start of 2000 to the start of year x as a function of x. (Do not simplify the answer.) HINT  [Rate of revenue = p(t)r(t).]

2. If f(x) = 1+ x2, find fof . * %3D A. 1 + x2 + x4 B. 2 + 2x2 + x4 C. 2 + x2 D. 1 + x4