3.7, Graphing Calculator Fun. Sketch a full picture of the following graphs. Draw axes and indicate the x and y max and min.

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1 3.7, Graphing Calculator Fun Sketch a full picture of the following graphs. Draw axes and indicate the x and y max and min. y = x 2 32x +240 y = -2sin(x) +1 [-2π, 2π] Solve the following 3x 2 4x 2 = 0 - x 2 24x = -20 Find all, if any, points of intersections y = x 4 3x y = sin(2x) y = -2x 3 x 5 y = -3cos(x) - 1

2 3.7, nderiv Math 8 nderiv( nderiv finds the of a derivative at a. f(x) = 6x 2 3x find f '(3). y = x 1 find y '(1) 3 y x 2 find y '(0) f(x) = tan(x) find f '(π/2) The Ti-83 and 84 do not find the answer by finding the and plugging in an. What these calculators use is arithmetic and the symmetric quotient: So as we have seen, we must be than the calculator and know some and leave the to the calculator. dy/dx Y 4 = nderiv( Y 5 = nderiv(

3 3.7, Optimization Remember that when a function is at a max or min its derivative equals Steps to follow to solve the following problems. 1. Draw a and label. 2. Find/create the, in variable, of what we are trying to find the or of. 3. Find the of that. 4. Set the equal to and and. 5. Keep in mind the of possible answers when solving in step 4. (a.k.a the ) Note: steps 3 and 4 can be done on the calculator but you must be explicit with your work shown. A rectangular piece of aluminum, 12 X 18 is to be folded up on each end to create a tray. The corners of the rectangle will be wasted, cut-off, in this folding. How many inches should the rectangle be folded up in order to create a tray that can contain maximum volume?

4 3.7, The x and y axes are the two legs of a right triangle drawn in the first quadrant. The hypotenuse of the triangle must pass through the point (1, 8). What is the minimum length of such a hypotenuse? Homework Page , 17, 23 A soda can is a right cylinder. A volume of 12 ounces is approximately equal to 22 cu.in. Find the dimensions of the can that would minimize total surface area, i.e. the amount of material needed to make such a can. Homework Page , 19c, 22, 25, 29

5 3.7, One day I went on an adventure. I drove my car to a parking lot on the edge of a park. The parking lot was (perpendicularly) 2 miles from the bay. I walked through the park, to the bay, and found a row boat. I got in the row boat and rowed around in the bay until I found myself at a point in the bay which was 0.5 miles from the shoreline, and 3 miles down the shore line from the point perpendicular to where I parked my car. At that moment I remembered that I had a dentist appointment. The appointment was for 2:30 p.m., and it was now 1:00 p.m. I wanted so badly to get my teeth cleaned that I began to try and determine how I could get to my car, and then the dentist, as quick as possible. I knew I could row at a rate of 10mph and jog to my car at a rate of 6 mph. I also knew I could take any path I needed to get from the shoreline to my car. And, I also also knew that the drive from my car to the dentist s office is 45 minutes. So help me please. a) Where on the shoreline should I row to, so that I can get to my car in as little time as possible, and hence to the dentist s office on time, if possible. b) What is my earliest possible arrival time at the office? Will I be able to keep my appointment?

6 3.7, Motion on a line Something traveling and forth, passing back over from it came, like a point traveling on an axis. A point is traveling on the x-axis in such a way that its position on the x-axis, x(t), at time t (in seconds), can be found using the function x(t) = t 3 12t + 2 on the interval t : [0, 5]. 1. Where is the object when t = 0, t = 1? 2. Find the velocity of the object a t = Find the speed of the object at t = Find the acceleration of the object at t = When is the object moving to the left? 6. When, if ever, is the object not moving? 7. When, if ever, is the object turning around? 8. At t = 1 is the object speeding up or slowing down. 9. Sketch the movement of this point. Homework Linear motion worksheet

7 3.7, f(x) = x x 0 = 1 x 1 = 2 Δx = Δy = Δx and Δy are used for measuring change on a function. Relate this to change at a point. Differentials

8 3.7, Linear Approximations A linear approximation is really just a - on the tangent of a graph, drawn from a point of tangency the indicated x value. Usually the point of tangency is a number. Find the linear approximation of f(1.01) if f(x) = x (two approaches) I guess that a quadratic approximation would be a Using a linear approximation, approximate Homework Page f(2.01), 2 f(1.9), 5 f(2.1),