Now here is the explanation of the question done in class. If needed depending on what the question asks, Plug the value of that variable into the one or (two) equations to find all dimensions. (First Derivative Test or Second derivative Test)Ħ. Set the derivative to zero, and solve for the remaining values. Write the derivative of the equation in the two variable form.ĥ. Try to get the equation into a two variable form, where you are finding one of the variables, because you want to find the derivative.Ĥ. Write an equation for it, make sure to use descriptive variables so you won't get confused when doing the problem.ģ. Find what you are trying to either maximize or minimize.Ģ. K gave the class the 6 rules to live by when doing optimization problems, although do not really live by them, just follow them loosely.ġ. although very rarely is the function given so you as a calculus student must use all your knowledge you've learnt up to now and create a formula/function that can be used to answer the question that is being asked. Optimization problems are problems that deal with either finding the maximum or minimum of the function in the problem. K introduced a new topic Optimization problems. We will be continuing optimization problems on Monday.Well i am finally back to scribe, it has been a pretty long time since i had last scribed so here it is.Slide #7 has a question for you to solve.Since we know t=1/29 hr, we put that into d to determine the minimum distance at 10:02am. (Who wants to use chain rule 20 times for deriving an equation? I don't.) We did this so we can find the critical number(s), determine if the critical number(s) is a max or min using the 1st derivative test, then solve for t.Īnswer: t=1/29 hr, which is approx, 2 min, so at 10:02am the cars are closest.Ĥ. (Someone, help me out here.) Then we expand the equation we generated for d, so we can apply the derivative rules as minimum as possible. Because d is a radical function and we're looking for the critical numbers, uh oh, I forgot. Yes, that's my adjective for hypotenuse it's been copyrighted by none other than me, zeph.ģ. Using the chart, we were able to generate a formula for d, the distance or the hypotenuse or the hypotenusal distance. Using the info provided, we organized the info onto a chart, where d=distance, r=velocity (a.k.a. Designated the hypotenuse of the triangle, d.Ģ.The derivative of the distance function is the velocity function. We can write designate the distance as a function of time.Car B is moving 50mph south, north of P.Find the time at which they are closest to each other and approximate the minimum distance between the automobiles.ġ. At that same instant another car is 2 miles north of P, traveling south at 50 mph. A car crosses P at 10:00 a.m., traveling east at a constant speed of 20 mph. This is the first derivative test.Ī North-South highway intersects an East-West highway at a point P. We know this by plugging a number 3 into the equation of A'. We realized that to the left of h=3 on the graph of A', the function is positive, and to the right, negative. If applicable, draw a figure and label all variables. The highest possible input of the graph would be 6, which would also give us an area of 0 (imagine two walls stuck together).Ĥ. Problem-Solving Strategy: Solving Optimization Problems. The lowest possible input of the graph would be 0, which would give us an area of 0 (imagine two walls stuck together). We then determined the domain of which the function can exist using the context in the problem. H = 3.Determine if the critical numbers are a maximum or a minimum using the 1st derivative test.ģ. From the equation that we set up, we can see that the area is a function of height.Ī'(h)=12-4h.Differentiated the previous equation so that we may find the critical numbers by finding the x-intercepts of the derivative of A. We also realized that this is an area problem, not a volume problem.Ī=(12-2h)h.The definition of the base is given in terms of height. We realized that the b=12-2h, since the rain gutter takes the shape of an open rectangle with only two sides.Ģ. Notes on Calculus and Optimization 1 Basic Calculus 1.1 Denition of a Derivative Let f(x) be some function of x, then the derivative of f, if it exists, is given by the following limit df(x) dx lim h0 f(x+h)f(x) h (Denition of Derivative) although often this denition is hard to apply directly. We drew the diagram and assigned the variables, base (b) and height (h). How many inches should be turned up to give the gutter its greatest capacity?ġ.
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