Aim: So, what is it this calculus thing can really do to solve problems? Do Now: ….. greatest profit ….. least cost ….. largest ….. smallest Write a function based on two equations.

Finding Minimum since C is a quadratic this 2nd D. is always + A manufacturing company has determined that the total cost of producing an item can be determined from the equation C = 8x2 – 176x + 1800, where x is the number of units that the company makes. How many units should the company manufacture in order to minimize the cost? Find critical values of x Looking for minimum: 2nd D. must be > 0 since C is a quadratic this 2nd D. is always + Must manufacture 11 units to min. costs

since h(t) is a quadratic this 2nd D. is always - Finding Maximum A rocket is fired into the air, and its height in meters at any given time t can be calculated using the formula h(t) = 1600 + 196t – 4.9t2. Find the maximum height of the rocket and at which it occurs. Find critical values of x Looking for maximum: 2nd D. must be < 0 since h(t) is a quadratic this 2nd D. is always - h(20) = 1600 + 196(20) – 4.9(20)2 = 3560 m.

Finding Maximum Volume A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? x h maximize volume primary equation – contains the quantity to be optimized Surf. Area = (area of base) + area of 4 sides = 180 secondary equation

Finding Maximum Volume A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? x h = 108 1. maximize Volume 2. express V as a function of one variable solve for h in terms of x replace h in primary equation

Finding Maximum Volume A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? domain for function is all reals, but . . . we must find the feasible domain x must be > 0 Area of base = x2 can’t be > 108 feasible domain

Finding Maximum Volume A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? 6 3 x h to maximize – find critical values = 0 x = ±6 evaluate V at endpoints of domain and 6

Problem Solving Strategy 1. Assign symbols to all given quantities and quantities to be determined. Sketch 2. Write a primary equation for the quantity that is to be optimized. 3. Reduce primary equation to one having a single independent variable. This may involve the use of a secondary equation relating the independent variables of the primary equation. 4. Determine the feasible domain of the primary equation. 5. Use calculus to optimize

Finding Minimum Distance Which points on the graph of y = 4 – x2 are the closest to the point (0, 2)? (x, y) (0, 2) primary equation y = 4 – x2 secondary equation

Finding Minimum Distance Which points on the graph of y = 4 – x2 are closest to the point (0, 2)? primary equation (x, y) secondary equation (0, 2) y = 4 – x2 rewrite w/one independent d is smallest when radicand is smallest f(x) = x4 – 3x2 + 4 = 0

Finding Minimum Distance Which points on the graph of y = 4 – x2 are closest to the point (0, 2)? f(x) = x4 – 3x2 + 4 = 0 f’(x) = 4x3 – 6x = 0 (x, y) find critical numbers (0, 2) x = 0 is a relative maximum evaluate y = 4 – x2 for x = min. distance from (0, 2)

rewrite w/one independent Model Problem Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area? Maximize what? A of + A of primary equation secondary equation solve for r: rewrite w/one independent

Model Problem feasible domain? 0 < x < 1 Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area? feasible domain? 0 < x < 1

maximum area occurs at x = 0 max when all wire is used for circle! Model Problem Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area? 0 < x < 1 domain find critical numbers is only critical value in domain evaluate primary equation A(0) 1.273 A(.56) .56 A(1)= 1 maximum area occurs at x = 0 + 0 max when all wire is used for circle!

Model Problem You are planning to close off a corner of the first quadrant with a line segment 15 units long running from (x, 0) to (0, y). Show that the area of the triangle enclosed by the segment is largest when x = y.

Model Problem Find the points on the hyperbola x2 – y2 = 2 closest to the point (0, 1).

Aim: So, what is it this calculus thing can really do to solve problems? Do Now: A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?