Chapter 1.1, Problem 1E

2) Compute the following anti-derivative. √1x4 dx

Question 3 (5pt): A chemical reaction. In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C : ABC. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: d[C] dt = k[A][B] (where k is a constant positive number). Thus, if the initial concentrations are [A] = = a moles/L and [B] = b moles/L we write x = [C], then we have (E): dx dt = k(ax)(b-x) 1 (a) Write the differential equation (E) with separate variables, i.e. of the form f(x)dx = g(t)dt. (b) Assume first that a b. Show that 1 1 1 1 = (a - x) (b - x) - a) a - x b - x b) (c) Find an antiderivative for the function f(x) = (a-x) (b-x) using the previous question. (d) Solve the differentiel equation (E), i.e. find x as a function of t. Use the fact that the initial concentration of C is 0. (e) Now assume that a = b. Find x(t) assuming that a = b. How does this expression for x(t) simplify if it is known that [C] =…

3) Find the volume of the solid that lies inside both the sphere x² + y² + z² cylinder x²+y² = 1. = 4 and the

1) Compute the following limit. lim x-0 2 cos(x) 2x² - x4

Question 1 (1pt). The graph below shows the velocity (in m/s) of an electric autonomous vehicle moving along a straight track. At t = 0 the vehicle is at the charging station. 1 8 10 12 0 2 4 6 (a) How far is the vehicle from the charging station when t = 2, 4, 6, 8, 10, 12? (b) At what times is the vehicle farthest from the charging station? (c) What is the total distance traveled by the vehicle?

Question 2 (1pt). Evaluate the following (definite and indefinite) integrals (a) / (e² + ½) dx (b) S (3u 2)(u+1)du (c) [ cos³ (9) sin(9)do .3 (d) L³ (₂ + 1 dz

= Question 4 (5pt): The Orchard Problem. Below is the graph y f(t) of the annual harvest (assumed continuous) in kg/year from my cranapple orchard t years after planting. The trees take about 25 years to get established, and from that point on, for the next 25 years, they give a fairly good yield. But after 50 years, age and disease are taking their toll, and the annual yield is falling off. 40 35 30 。 ៣៩ ថា8 8 8 8 6 25 20 15 10 y 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 The orchard problem is this: when should the orchard be cut down and re- planted, thus starting the cycle again? What you want to do is to maximize your average harvest per year over a full cycle. Of course there are costs to cutting the orchard down and replanting, but it turns out that we can ignore these. The first cost is the time it takes to cut the trees down and replant but we assume that this can effectively be done in a week, and the loss of time is negligible. Secondly there is the cost of the labour to cut…