Sample Lecture Outline for Fall Semester MATH 1350

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1 Sample Lecture Outline for Fall Semester MATH 1350 Before semester starts: Prep computer graphing utility for Day 4 Prep computer secant tangent grapher for Day 8 (will involve getting permission for JAVA applet) Day 1 X Discuss Course Organization Today: Section 21 Limits: Graphical Approach The Definition of limit lim f x L symbol: x c spoken: The limit, as x approaches c, of f(x) is L less-abbreviated symbol: f(x) L as x c spoken: f(x) approaches L as x approaches c usage: x is a variable, f is a function, c is a real number constant, and L is a real number constant meaning: as x gets closer & closer to c, but not equal to c, the value of f(x) gets closer & closer to L (may actually equal L) Leave blank line to be filled in later today we will do graphical approach Examples of graph description of limit behavior [Class Drill 1] Limits project the class drill on one screen, then project the book on the other screen and show that the class drill is similar to suggested problems 3-1#7,8,13,15 do row x = 1 Fill in the blank line in the definition of limit: Graphical Interpretation: The graph of f appears to be heading for location (x, y) = (c, L) Do row x = 4 Point out the difference between the idea of the existence of a y-value at x=a and the existence of the limit as x a do row x = -1 Define one-sided limits Re-cast the definition of limit using 3-part test involving one-sided limits finish row x = -1 do row x=-3 Now: Examples of description of limit behavior graph [Example 1] IMPORTANT: Exercise 3-1#40 Sketch a graph that satisfies all these conditions: f(1) = 3 lim f(x) = 2 x 1 lim x 1 + f(x) = 4

2 Day 2 X Continuing Section 21 Limits Analytical Approach Today: Analytical Approach to Limits So far, we have done graphical approach to limits Now do Analytical Approach Project Reference 2: Facts about Limits from Section 3-1 [Example 1] Let f(x) = 7x x 25 (A) Find f( 2) (B) Find lim x 2 f(x) [Eample 2] Let f(x) = 24 + x 2 (A) Find f(5) (B) Find lim x 5 f(x) Examples involving cancelling inside the limit ( algebraic simplification ) [Example 3] using rational function f(x) = x2 2x 3 x 3 = (x 3)(x+1) (x 3) (A) Find f(c) for c=1,2,3 Note that we cannot cancel 0/0 So f(2) DNE (B) Find lim x c f(x) for c = 1,2,3 Note most important step: we can cancel (x 3)/(x 3) Explain why we can cancel: In all three cases c = 1,2,3, we know that x 3 (explain why) This tells us that x 3 0, so we can cancel (x 3)/(x 3) common mistake: lim f(x) = lim = 0 DNE (one mistake on this line) x 3 0 another: lim f(x) = lim = (3 3)(3+1) = (3 + 1) = 4 (two mistakes on this line!) x 3 (x 3) (3 3) Discuss why this example makes sense Let g(x) = x + 1 Make side-by-side graphs of f(x) and g(x) on separate axes Using the graphs, explain why the answers in question (A),(B) make sense (draw lines like those in the drawing on page 130, Example 2, figure 2) The functions f(x) and g(x) are not the same function, but they do have the same limit Discuss hole in graph at (x, y) = (3,4) It corresponds to the fact that f(3) DNE, but lim f(x) = 4 x 3 (x 3)(x+1) x 3 (x 3) (x 3)(x+1) x 3 = (3 3)(3+1) (3 3) [Example 4] IMPORTANT using rational function f(x) = x2 6x+5 = (x 1)(x 5) x 2 8x+15 (x 3)(x 5) (A) Find f(c) for c = 5,3 (B) Find lim f(x) for c = 5,3 x c IMPORTANT Notice that Theorem 4 on page 103 tells us that limit lim f(x) DNE x 3 Notice: at x = 5: f(5)dne while lim f(x) = 2 So graph has a hole at (x, y) = (5,2) caused by (x 5) x 5 (x 5) 1 at x = 3: f(a)dne and lim Don t yet know what this causes in graph f(x) DNE because x 3 (x 3) IMPORTANT Difference Quotient examplesfrom Section 21 [Example 5] (Similar to to Suggested Exercise 21#67): For f(x) = x 2 6x + 5 f(4+h) f(4) Find lim = 2 difference quotient h 0 h Examples involving piecewise-defined functions (Leave them to read this in book) 2x + 10, x 3 [Example 6] similar to #45 Let f(x) = { x 2, x > 3 (A) Find lim f(x) (B) Find f(3) (C) Explain with a graph x 3 [Example 7] similar to #49 Let f(x) = x 11 (A) Find f(11) x 11 (B) Find lim f(x) (C) Find lim f(x) (D) Find lim f(x) (E) Explain with a graph x 11 + x 11 x 11

3 Day 3 X Section 22 Limits Involving Infinity Introduce Limits Involving Infinity Define infinite limits symbol: lim f(x) = x c spoken: The limit, as x approaches c, of f(x), is infinity Meaning: as x gets closer & closer to c but not equal to c, the y-values get more & more positive without bound Graphical Significance: Graph of f has a vertical asymptote at x = c Note that the line equation for the asymptote is x = c Remark: Important point: we have revised the definition of limit! In Section 22, the book says the limit does not exist, but we write the symbol lim f(x) = That s nonsense We have x 3 changed the definition of limit With the old definition of limit (from section 21), the limit does not exist With the new definition of limit (from section 22), the limit does exist and it is infinity Obvious variations: lim x c + f(x) = or lim x c f(x) =, etc Ask the students to explain Define limits at infinity symbol: lim f(x) = b x spoken: The limit, as x goes to infinity, of f(x), is b Meaning: as x gets more & more positive without bound, the y-values get closer & closer to b Graphical Significance: Graph of f has a horizontal asymptote on the right at y = b Note that the line equation for the asymptote is y = b Obvious variation: lim x f(x) = b or lim x Another possibility for limit at infinity: lim x f(x) = Ask the students to explain f(x) = DNE Sketch example Limits Involving Infinity for a Function Given by a Graph [Class Drill 2] Limits Involving Infinity for a Function Given by a Graph Remark: Similar to Suggested Exercises 22#9-16, but not covered in any book examples! Limits Involving Infinity for a Function Given by a Formula [Class Drill 3] Guessing Limits by Substituting in Numbers Remarks about Class Drill 3 Note: f(x) is the same function as Day 2 Example 4 Book often analyzes this kind of function with a calculator, but we can do it by hand At x = 1: The y-value f(1) = 0, point on graph graph at (x, y) = (1,0) This is an x-intercept At x = 5: The y-value f(5) DNE but lim f(x) = 2 So hole in graph at (x, y) = (5,2) x 7 At x = 3:The y-value f(3) DNE According to Section 21 Thm 4, the lim f(x) DNE as well x 3 But as x approaches 3 from the right, the y-values get more&more positive without bound And as x approaches 3 from the left, the y-values get more&more negative without bound Using the terminology of infinity: lim f(x) = and lim f(x) = but still lim f(x) DNE x 3 + x 3 x 3 Observations about Factors Factored form of function gives us the x-values where important features of the graph occur Factored form is also the best form for determining the precise behavior at those x-values Factors of form (x c) in numerator alone cause an x-intercept in the graph at (x, y) = (c, 0) Factors of form (x c) in numerator and denominator with equal powers cause a hole in the graph (x c) at x = cthe y-coordinate of the hole is the real number lim f(x) x c Factors of form 1 (x c) in denominator alone cause a vertical asymptote in the graph at x = c

4 Day 4 PREP COMPUTER GRAPHER FOR THIS DAY X Continuing Section 22 Limits Involving Infinity Today: Limits at Infinity and Horizontal asymptotes [Example 1] Let f(x) = 7x2 42x+35 = 7(x2 6x+5) = 7(x 1)(x 5) 2x 2 16x+30 2(x 2 8x+15) 2(x 3)(x 5) (Note: f(x) is the same function as Day 2 Example 4 and Class Drill 3, but multiplied by 7/2) Consider end behavior That is, what happens as x? Take lim f(x) and see what happens x Using technique of identifying dominant terms, we find lim f(x) = 7 x 2 That is, as x values get more and more positive without bound, the y-values get closer to y = 7 2 Get computer graph Note that graph of f has a horizontal asymptote on the right at y = 7 2 [Example 2] Let g(x) = 7x2 42x+35 2x 3 16x 2 +30x = 7(x2 6x+5) 2x(x 2 8x+15) = 7(x 1)(x 5) 2x(x 3)(x 5) Now find lim x g(x) Using technique of identifying dominant terms, we find lim x g(x) = 0 That is, as x values get more and more positive without bound, the y-values get closer and closer to y = 0 Get computer graph Note that graph of g has a horizontal asymptote on the right at y = 0 [Example 3] Let h(x) = 7x3 42x 2 +35x 2x 2 16x+30 = 7x(x2 6x+5) = 7x(x 1)(x 5) 2(x 2 8x+15) 2(x 3)(x 5) Now find lim h(x) x Using technique of identifying dominant terms, we find that as x values get more and more positive without bound, the y-values get larger and larger positive, without bound That is, lim h(x) = x Get computer graph Note that graph of h goes up on right Conclusion we have observed that The standard form of rational function is most useful for determining behavior as x If lim f(x) = b, then the graph of f has a horizontal asymptote on the right at y = b x That is, the line equation of the asymptote is y = b If lim x f(x) =, then the graph of f goes up on the right There is no horizontal asymptote on the right Obvious question: are these the only possibilities? Could have lim x f(x) =, then the graph of f goes down on the right Could have lim x f(x) = DNE then the graph of f does not have simple behavior on the right Also notice: In our graphs of f and g, there was also a horizontal asymptote on the left with the same line equation This is a fact about Rational Functions If f is a rational function and if f has a horizontal asymptote on the right, then the graph will also have a horizontal asymptote on the left at the same height This is not true of general functions that are not rational functions Example Exponential Also if a graph of a rational function goes up on one end, it can go up or down on the other end Example y = x 2 or y = x 3 But the graph cannot have a horizontal asymptote on other end Quiz 1

5 Day 5 X Continuing Section 22 Limits Involving Infinity, continued Today: More examples of limits involving infinity for Rational Functions Applications of Limits at Infinity: Drug Concentration Problems [Example 1] (similar to Suggested Exercise 22#89) Drug administered to a patient through an injection The drug concentration in the bloodstream is described by the function C(t) = 5t2 (t + 50) t where t is the time in hours after the injection and C(t) is the drug concentration in the bloodstream (in milligrams/milliliter) at time t Find and interpret lim C(t) t [Example 2] (similar to Suggested Exercise 22#89) Drug administered to a patient through an IV drip The drug concentration in the bloodstream is described by the function C(t) = 5t(t + 50) t where t is the time in hours after the drip was started and C(t) is the drug concentration in the bloodstream (in milligrams/milliliter) at time t Find and interpret lim C(t) t Question: do these two problems make sense? Locating asymptotes [Example 3] (horiz asymp, vert asymptote and hole) (Similar to Suggested Exercise 22#63) Find all horizontal and vertical asymptotes for the function: f(x) = (3x2 3x 36) (2x 2 6x 8) = 3(x2 x 12) 3(x + 3)(x 4) 2(x 2 = 3x 4) 2(x + 1)(x 4) Do more than just that Explain effect of each factor on the graph behavior Use limit terminology where appropriate Explain end behavior using limit terminology [Example 4] (horiz asympt, no vert asympt or hole) Same Instructions as [Example 1] g(x) = (3x2 3x 36) (2x 2 6x + 8) = 3(x2 x 12) 3(x + 3)(x 4) 2(x 2 = 3x + 4) 2(x 2 3x + 4) Summary of analytic approach to limits involving infinity The standard form of a rational function f(x) is useful for finding lim x f(x) to find horizontal asymptotes and determine end behavior The limit is found by identify the leading terms in the numerator & denominator and analyzing their degrees and leading coefficeints The factored form of f(x) is useful for finding x-intercepts, vertical asymptotes, holes Recall observations about factors from Day #3 notes

6 Day 6 X Section 23: Continuity Return to Class Drill 1 about limits Observe that to draw graph at holes & jumps, we have to lift our pen The concept of continuity describes this distinction in abstract mathematical terms Definition of Continuity (formal and informal) do [Class Drill 4] Limits and Continuity Sign behavior of functions Define continuous on an interval Observations Point out that polynomial functions are always continuous So for polynomial functions, the equation lim x a f(x) = f(a) is always true This is where Limit Theorem 3 came from! Point out that rational functions are continuous everywhere except at x-values that cause the denominator to be zero (These x-values can be the locations of holes or of vertical asymptotes Both are types of discontinuities) So for rational functions, the equation lim x a f(x) = f(a) is always true as long as x=a is in the domain This is where the other part of Limit Theorem 3 came from! Point out that a function can only change sign by touching the x-axis and crossing it (an x- intercept) or by jumping across the x-axis (an x-value where the function is discontinuous) Define partition numbers to be x-values where the y-value is zero or where the function is discontinuous This gives us an insight into sign behavior: the sign behavior is unchanged on each interval between partition numbers using sign behavior to solve inequalities [Example 1] for f(x) = 9x 2 90x = 9(x 3)(x 7) (a) Determine sign behavior of f Point out difficulty of using sample numbers (b) solve f(x) 0 present answer 3 ways [Example 2] Solve 9x 2 90x 189 Present answer three ways [Example 3] Solve x 3 49x Present answer three ways Incorrect and correct solution Day 7 X continuing Section 23: Studying the sign behavior of functions [Example 1] f(x) = x2 5x (a) Determine sign behavior of f Point out difficulty of using sample x 4 numbers (b) solve f(x) 0 present answer 3 ways [Example 2] f(x) = (3x2 3x 36) (2x 2 6x 8) = 3(x2 x 12) 3(x + 3)(x 4) 2(x 2 = 3x 4) 2(x + 1)(x 4) Solve inequality f(x) 0 Present answer 3 ways [Example 3] Return to Day 5 [Example 1] (Similar to 22#63) Find all horiz and vert asympt: f(x) = (3x2 3x 36) (2x 2 6x 8) = 3(x2 x 12) 3(x + 3)(x 4) 2(x 2 = 3x 4) 2(x + 1)(x 4) Do more than just that Explain effect of each factor on the graph behavior Use limit terminology where appropriate Explain end behavior using limit terminology Observe: Now that we have a methodical way to anayze sign behavior, we have an easier way to determine limit behavior at vertical asymptotes Quiz 2

7 Day 8 PREP WEBSITE FOR THIS DAY!!! X Section 24 The Derivative Today: Rates of Change We will do a series of examples involving f(x) = x x 16 = (x 2)(x 8) (A) Draw the graph on paper (B) Draw the secant line that passes through the points (3,f(3)) and (5,(f(5)) (C) Find slope of that secant line Result: using m = Δy, we get m = 2 Confirm using Secant Tangent grapher at the following website: PREP WEBSITE!! Introduce Average Rate of Change Project Reference 3 Rates of Change (p3 of course packet) on the movie screen Discuss the definition of Average Rate of Change, and the fact that the secant line slope calculation that we just did was an example of a calculation of an average rate of change That is, the average rate of change of f(x) = x x 16 from x = 3 to x = 5 is number m = 2 (D) Find avg rate of change of f(x) = x x 16 from x = 3 to x = 3 + h where h 0 Result: m = f(3+h) f(3) h = 4 h (Very important can cancel h/h because h 0) (E) Make a new graph of f and Illustrate this quantity on that new graph Introduce the Tangent Line (F) Draw a new graph and Draw the line tangent to the graph of f at x = 3 Goal: Find the slope of that tangent line That is, find the slope m of the line tangent to the graph of f(x) = x x 16 at x = 3 Discuss the fact that we don t have two known points on the tangent line, so we can t use our slope formula m = Δy to find the slope of the tangent line Δx View computer graph on Secant Tangent Grapher Turn on Tangent line Observe that by bringing right intersection point of Secant Line closer to left intersection point, the secant line changes to look more & more like tangent line Observe that the slope of secant line seems to be getting closer and closer to the number m=4, and it seems that this number m=4 is probably the slope of the tangent line Analytically, pulling the second point closer to the first corresponds to finding the limit f(3 + h) f(3) m = lim h 0 Do that limit for our computation We find m = lim Δx h f(3+h) f(3) h 0 h = lim h = 4 Official introduction of the tangent line The line tangent to the graph of f at x=a is defined to be the line that has these two properties: (1) The line touches the graph of f at x=a This means that the point (x,y)=(a,f(a)) is on the line This point is called the point of tangency (Note that this requires that f(a) must exist) f(x+h) f(x) (2) The line has slope m = lim if this limit exists and is a real number h 0 h Graphical interpretation: The tangent line (if it exists) is the line that passes through the point of tangency (x,y)=(a,f(a)) and looks like it is going the same direction as the graph of f at that point [Class Drill 5] Representations of Slopes

8 Day 9 X continuing Section 2-4 Introduce the idea of the derivative function, f Imagine a machine: input: number a Instructions inside machine: (find slope of the line tangent to graph of f at x=a) output obtained graphically: number m that is slope of line tangent to the graph of f at x=a output obtained analytically: the number m = f (a) = lim f(a+h) f(a) h 0 h This machine is really called the derivative machine Discuss using x instead of a [Class Drill 6] Finding derivatives graphically using a ruler Examples of computing f (x) using analytic definition of the derivative: f f(x+h) f(x) (x) = lim h 0 h [Example 1] (Sim to Sugg Exercise 24#27) f(x) = x 2 2x 3 Find f (x) analytically, using the definition of the derivative Write explanation for most important step: Cancelling h/h Result: f (x) = 2x 2 Observe that f(x) = x 2 2x 3 and f (x) = 2x 2 match the graphs in class drill 6! Related questions (A) Find the slope of the line that is tangent to the graph of f at x=2 (B) Find the slope of the line that is tangent to the graph of f at x=0 (C) Find the x-coordinates of all points on the graph of f that have horizontal tangent lines [Example 2] (Sim to Sugg Exercise 24#27) Find derivative of function f(x) = 3x 2 + 5x 7 Day 10 X continuing Section 24 The Derivative Today: More examples [Example 1] f(x) = 1/x Find the Derivative of f That is, find f (x) using the def of deriv [Example 2] (Sim to Sugg Exercise 24#33) harder example involving a 1/x-type function: Let f(x) = 3 7 Find f (x) x [Example 3] f(x) = x Find the Derivative of f That is, find f (x) using the def of deriv [Example 4] (Sim to Sugg Exercise 24#35) harder example involving a x-type function: Let f(x) = 3 7 x Find f (x) Nonexistence of the Derivative Draw graph with common pathologies: Missing point Jump in Graph Cusp Vertical tangent Discuss why the derivative does not exist at each point Remark that both Cusp Point and Vertical Tangent are locations where the function exists, the limit exists, and the function is continuous Only with the concept of derivative are we finally able to articulate what is bad about these locations

9 Day 11 X Section 25 Basic Differentiation Properties Today: Constant Function Rule; Power Rule Review Def of deriv In Section 25, we will learn rules for taking derivatives We won t use def Notation for Derivatives The Constant Function Rule Present the Rule [Example 1]: If f(x) = 7 then f (x) = 0 That is, d 7 = 0 Does this make sense graphically? dx The power rule Present the power rule Two equation form: If f(x) = x n, then f (x) = nx n 1 Single Equation form: d dx xn = nx n 1 [Example 2] f(x) = x 4 [Example 3] f(x) = 1 Compare to result of Day 10 [Example 1] using definition of derivative x [Example 4] f(x) = x Compare to result of Day 10 [Example 3] using definition of derivative Discuss importance of step 1: rewriting function then step 2: take derivative Discuss common incorrect notation [Example 5] f(x) = x Use Power Rule Then check to see if it makes sense graphically [Example 6] f(x) = 1 Compare to result obtained using Constant Function Rule Quiz 3

10 Day 12 X continuing Section 25 Basic Differentiation Properties, continued Today: The Sum and Constant Multiple Rule Present the rule [Example 1] Find deriv of f(x) = 3x 2 + 5x 7 Present method that I want them to be able to do Step 1: rewrite f as sum of terms of form constant*power function Step 2: identify multiplicative constants and use sum & const multiple rule Simplify result to eliminate negative exponents Compare to result of Day 9 Example 2 using definition of derivative [Example 2] Find derivative of f(x) = 3 7 x Compare to result of Day 10 [Example 2] using definition of derivative [Example 3]Find derivative of f(x) = 3 7 x Compare to result of Day 10 [Example 4] using definition of derivative [Class Drill 7] Find deriv of f(x) = 7 x Trick Problems x 2/5 [Example 4] Trick problem (Sim to Sugg Exercise 25#81): Let f(x) = 2x5 4x 3 +2x x 3 Find f (x) Special Topics The Equation for the Tangent Line Review point-slope form (y b) = m(x a) Remember what we know about the line tangent to graph of f at x = a Analytical description: tangent line is the line that has these two properties: 1) Touches graph at x That point is called point of tangency The y-coord is f(a) So point of tangency has coordinates (x, y) = (a, f(a)) 2) Tangent line has slope m = f (a) This is the KNOWN SLOPE of the tangent line Point Slope form of equation for tangent line: (y f(a)) = f (a)(x a) [Example 5] tangent line problem (Similar to Suggested Exercise 25#59): Let f(x) = x 3 9x x + 25 = (x + 1)(x 5) 2 Find equation of the line tangent to graph at x = 2 [Class Drill 8] Questions about Tangent Lines Let f(x) = x 3 3x 2 9x + 11 (a) Find f (x) Use the techniques of Section 25 (That is, DO NOT use the Definition of the Derivative)Show all details clearly and use correct notation (b) Find the slope of the line that is tangent to the graph of f at x = 3 (c) Find the slope of the line that is tangent to the graph of f at x = 0 (d) Find the x-coordinates of all points on the graph of f that have horizontal tangent lines (e) Find the equation of the line that is tangent to the graph of f at x = 2 Show all details clearly and present your equation in slope intercept form (Remember that the approach is to build the general form of the equation for the tangent line in point-slope form (y f(a)) = f (a) (x a) and then convert the equation to slope-intercept form)

11 Day 13 X Section 27 Marginal Analysis Discuss Definition of Demand, Price, Revenue, Cost, Profit from Reference 5 Marginal Quantities Definition of Marginal Quantities from Reference 5 Examples involving Marginal Quantities For the coming 7 examples, use C(x) = x and R(x) = 5x 0 02x 2 [Example 1] (Similar to Suggested Exercise 27#9) Find the marginal cost function [Example 2] (Similar to Suggested Exercise 27#17) Find the marginal profit function Average Quantities OMIT AVERAGE QUANTITIES Estimation Problems [Example 3] (Similar to Suggested Exercise 27#33) The total cost of producing x electric guitars is C(x) = x 025x 2 dollars (A) What is the cost of producing a batch of 50 guitars? (B) What is the cost of producing a batch of 51 guitars? (C) If batch size changes from x=50 guitars to x=51 guitars, what will be change in the cost of producing a batch of guitars? That is, if x = 51 and Δx = 1, what is ΔC? (exact value) (Book wording: Find exact cost of producing the 51 st guitar ) Use Wolfram, result is ΔC = $7475 Idea of approximation: Exact Change is Δy = f(x 2 ) f(x 1 ) Approximate change is f (x 1 ) Δx important to display the relationship correctly, and to be clear about what one is presenting exact change = ΔQ approximate change = Q (x 1 ) Δx (D) If the batch size changes from x=50 guitars to x=51 guitars, use the marginal cost function to find an approximate value for the change in the cost of producing a batch of guitars That is, Use the marginal cost function to find an approximation for ΔC (Book wording: Use marginal cost to approximate the cost of producing the 51 st guitar ) Result C (x 1 ) Δx = 75 1 = 75 Compare approximate and Exact Results [Example 4] (Similar to Suggested Exercise 27#43) Price p (in dollars) and demand x for anitem are related by the equation x = p (A) Find price p in terms of x, and domain result: p = x + 50 Domain 0 x (B) Find the revenue R(x) from the sale of x items What is the domain of R? Result: R(x) = xp = x ( x x2 + 50) = + 50x Domain is 0 x (C) Find the marginal revenue at a production level of 400 items and interpret the results Result: R (x) = x + 50 So 10 R (400) = = = 10 Therefore if company 10 changes from batch size of 400 to batch size of 401, the revenue will increase by approximately $10 The book would say that this is approx the increase in revenue from sale of 401 st item (D) Find the marginal revenue at a production level of 650 items and interpret the results Result: R (650) = = = 15 Therefore if company changes from batch 10 size of 650 to batch size of 651, revenue will decrease by approx $15 The book would say that this is approximately the decrease in revenue from the sale of the 651 st item

12 Day 14 X Continuing Section 27 Marginal Analysis today: another example [Example 1] (Similar to Suggested Exercise 27#47) A company makes toasters Marketing department estimates a weekly demand of 300 toasters at price of $25 per toaster, and a weekly demand of 400 toasters at price of $20 per toaster Financial department estimates fixed weekly costs are $5000 and variable costs are $5 per toaster (A) Assume linear relationship between x and p Find equation that expresses p as a function of x, and find domain Result: We want equation of line through two known points (x, p) = (300,25) and (x, p) = (400,20) We find m = = 5 = 1 Use point-slope form: (p 25) = 1 (x 300) Convert to slope intercept form: p = 1 x The domain is 0 x 800 (B) Find revenue function in terms of x and state its domain Solution: : R(x) = xp = x ( 1 20 x + 40) = 1 20 x2 + 40x The domain is 0 x 800 (C) Assume cost function is linear Find cost function Solution: C(x) = 5x (D) Graph the cost function and revenue function on the same coordinate system for 0 x 800 Find the break-even points and indicate regions of loss and profit Solution: Downward-facing parabola, skewered by line Break-even points at x=200 and x=500 Corresponding values of price are p=6000 and p=7500 (E) Find Profit function Res: P(x) = ( 1 20 x2 + 40x) (5x ) = 1 20 x2 + 35x 5000 (F) Evaluate the marginal profit at x=325 and x=425 and interpret the results Solution: First we find the marginal profit function P (x) = 1 x + 35, 10 Using this, P (325) = 1 (325) + 35 = = 25 This tells us that if batch size 10 increases from 325 toasters per week to 326 per week, profit will increase by roughly $25 Using this, P (425) = 1 (425) + 35 = = 75 This tells us that if batch size 10 increases from 425 toasters per week to 426 per week, profit will decrease by roughly $75 Day 15 is Exam 1

13 Day 16 X Section 31: The constant e Interest formulas Start with Simple Interest Simple interest formula: A = P + Prt = P(1 + rt) Introduce P,r,t,A Graph equation [Example 1] Deposit $1000 into bank account with 3% simple interest What will the balance be after 7 years? Solution: A = 1000( ) = 1000(1 + 21) = 1000(121) = $1210 Periodically Compounded Interest Discuss the idea of compound interest Graph it Present general formula for Periodically Compound Interest: A = P (1 + r m )m t [Example 2] Deposit $1000 into bank account with 3% interest compounded yearly What will be the balance after 7 years? Show how it turns out A = 1000(1 + 03) 7 $ [Example 3] Deposit $1000 into bank account with 3% interest compounded monthly What will be the balance after 7 years? Show how it turns out A = 1000 ( )12 7 $ [Example 4] Deposit $1000 into bank account with 3% interest compounded daily What will be the balance after 7 years? Result: A = 1000 ( )365 7 $ More Frequent Compounding Observe: $1210 < $ < $ < $ for compounding (never, m=1,12,365) Question: What will happen to the value of a as m? That is, what is lim m P (1 + r m )m t?

14 Day 17 X Continuing Section 31 Today: The constant e and continually compounded interest Recall Question from Day 16: What will happen to value of A as m? That is, what is lim m P (1 + r m )m t? Related question: What is lim n (1 + 1 n )n? Investigate with table Explore values of (1 + 1 n )n as n y-values seem to be getting closer & closer to Based on this, we might guess that The limit lim (1 + 1 n n )n exists The value of the limit is near BIG FACTS FROM HIGHER MATH The limit lim (1 + 1 n n )n does exist The symbol e is used to denote the real number that is the value of the limit That is lim (1 + 1 n n )n = e, The number e is between 2 & 3 That is, 2 < e < 3 The value of e is near but e is irrational (e cannot be written exactly as a fraction or as a terminating decimal, or as a repeating decimal Related limit: lim (1 + x n n )n? Answer lim (1 + x n n )n = e x Discuss graph of y = 2 x, 3 x, e x Related limit: lim P (1 + r n n )nt? Answer lim P (1 + r n n )nt = Pe rt That answers our question from above Fact: lim m P (1 + r m )m t = Pe (rt) Discuss graph of y = Pe (rt) Inspired by this, invent a bank account that uses the formula A = Pe (rt) to compute its balance Call it continuously-compounded interest Add this to our lists of types of interest [Example 1] Deposit $1000 into bank account with 3% interest compounded Continuously What will be the balance after 7 years? Result: A $ Solve the equation A = Pe (rt) for t in terms of the others [Example 2] Deposit $937 into account with 23% interest compounded continuously What is the balance after 7 years?solution: A = Pe (rt) = 937e (7 0023) $ [Example 3] Deposit $937 into account with 23% interest compounded continuouslyhow long after initial deposit until the Balance has grown to 1200?Solution: t = ln( ) years [Example 4] Deposit some money into account with 23% interest compounded continuously How long until the balance doubles? [Example 5] If you want an account with continuously compounded interest to double in 20 years, what interest rate will you need?

15 Day 18 X Section 32 Derivatives of Exponential and Logarithmic Functions Today: Exponential functions New Rules New rule: Exponential Function Rule #1 d dx e(x) = e (x) This rule is proven using calculus above the level of this course Does this make sense graphically? New rule: Exponential Function Rule #2 d dx e(kx) = ce (kx) This rule is also proven using calculus above the level of this course The book introduces this fact only in exercises, in exercises 32 # 61, 62 The book never presents this in a list of derivative rules, never gives it a theorem number That is a shame, because it is one of the most-used derivative rules Use Rule #2 to prove Exponential Function Rule #3 d dx b(x) = b (x) ln(b) Review Derivative Rules so far in Reference 4 on page 4 of course packet [Class Drill 9] Find the derivatives of the following functions: (A) f(x) = 5e (x) (B) f(x) = 5e (7x) (C) f(x) = 5 7 (x) (D) f(x) = 5e (7) (E) f(x) = 5x e More difficult example involving exponential function Tangent Line Example [Example 1] (Similar to Suggested Exercise 32#29) For the function: f(x) = 11e (x) + 23x Find equation of line tangent to graph of f at x = 0 Find equation of line tangent to graph of f at x = 1 Rate of Change Examples [Example 2] Investment of $P earns interest at an interest rate of r compounded continuously (A) What is the value at t years (B) What is the instantaneous rate of change of the balance at time t years? [Example 3] (Similar to Suggested Exercise 32#71) Investment of $1000 earns interest at an annual rate of 2% compounded continuously (A) What is the value at 5 years? (B) What is the instantaneous rate of change of the balance at time 5 years? [Example 4] (Similar to Suggested Exercise 32#63) Value of a truck is given by the function S(t) = 174,000(09) (t) (A) What is the purchase price of the truck? (B) What is the value of the truck after 5 years? (C) What is the rate of change of the value of the truck at time t = 5 years? (D) Write a sentence summarizing the results of (B) and (C) That is, interpret them

16 Day 19 Fri Continuing Section 32 Derivatives of Exponential and Logarithmic Functions Today: Derivatives of Logarithmic Functions Review definition of logarithm function and shape of logarithm Graph New Derivative Rules Log Function Rule #1: d ln(x) = 1 Does this make sense graphically? dx x Log Function Rule #2: d log dx b(x) = 1 x ln(b) Review Derivative Rules so far in Reference 4 on page 4 of course packet [Examples]Find the derivatives of the following functions: (A) f(x) = 12 ln(x) (B) f(x) = 12 log 13 (x) (C) f(x) = 12 log(x) (D) f(x) = 12 ln(13) (E) f(x) = 12 ln(13x) [Class Drill 10] (A) Let f(x) = 12 ln ( 13 ) Find f (x) x (B) Let f(x) = 12 ln(x 13 ) Find f (x) (C) Let f(x) = 12x ln(13) Find f (x) (D) Find equation of line tangent to f(x) = 5 + ln(x 3 ) at x = e 2 Quiz 4

17 Day 20 X Section 33: Derivatives of Products and Quotients The product rule present the product rule Obvious thing: d (f(x)g(x)) = dx f (x)g (x) is WRONG! Present the correct rule: d dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) Review Derivative Rules so far in Reference 4 on page 4 of course packet [Example 1] (Similar to Suggested Exercise 33#21) f(x) = ( 3x 2 + 5x 7)(3x 2) Find f (x) using the product rule Simplify your answer [Example 2] (Similar to Suggested Exercise 33#55) Let f(x) = ( 3x 2 + 5x 7)e (x) (A) Find f (x) and simplify(b) f (0) (C) f (1) [Example 3] (Similar to Suggested Exercise 33#19) Let f(x) = 5x 7 ln(x) (A) find f (x) and simplify (B) f (1) (C) f (e) Day 21 X Continuing Section 33 Derivatives of Products and Quotients The Quotient rule present the quotient rule Obvious thing: d dx ( top(x) bottom(x) ) = top (x) bottom (x) is WRONG! Present the correct rule: d ( top(x) ) = top (x)bottom(x) top(x)bottom (x) dx bottom(x) (bottom(x)) 2 Review Derivative Rules so far in Reference 4 on page 4 of course packet [Example 1] (similar to Sugg Exercise 33#25) f(x) = (3x+5) Find f (x) and simplify answer (x 2 3) [Example 2] (similar to Sugg Exercise 33#31) f(x) = e(x) (x 2 3) Find f (x) and simplify answer [Class Drill 11] Don t forget the easy derivative rules (Section 33) Four parts! Trick Problems [Example 3] (similar to Suggested Exercise 33#73) Let f(x) = x4 +4 The goal is to find f (x) and simplify, by two methods x 4 (A) use quotient rule and then simplify (B) first simplify f(x) then find f (x) [Example 4] (similar to Suggested Exercise 33#87) Let f(x) = 3x5 2x 7 5 Find f (x) Discuss only: Don t do problem x 3

18 Day 22 X Continuing Section 33 Derivatives of Products and Quotients today: More difficult Quotient Rule problems Tangent Line Problems [Example 1] (sim to Sugg Ex 33#65): f(x) = 2x (x) Find equation of line tangent to f at x = 3 [Example 2] (similar to Sugg Exercise 33#69) Let f(x) = x x (A) find f (x) (B) find f (0) (C) find the x-coordinate of all the points on the graph of f that have horizontal tangent lines Approximation Problem [Example 3] (similar to Sugg Exercise 33#93) Sales of a game are described by the function S(t) = 7000t t+6 where t is the time (in months) since the game was introduced and S(t) is the total sales at time t (A) Find S(4) (B) Find S (t) Show all details clearly, use correct notation, and simplify your answer (C) Find S (4) (D) Write a brief interpretation of the answers from (A) and (C) That is, explain what the answers tell us Include the correct units in your explanation (E) Use the results of (D) to estimate the total sales after 5 months (F) How many games will eventually sell? Day 23 X Section 34 The Chain Rule introduce the chain rule, used for taking derivatives of nested functions Functions of form outer(inner(x)) The Chain Rule: d dx outer(inner(x)) = outer (inner(x)) inner (x) Review Derivative Rules so far in Reference 4 on page 4 of course packet Today: examples where the outer function is a power function Book solves such problems using what it calls the General Power Rule I won t call it that We ll just use the chain rule [Example 1] (similar to Sugg Exercise 34#21) For f(x) = 2(3x 4 + 5x 2 + 6) 7 find f (x) 2 [Example 2] (similar to Sugg Exercise 34#59) For f(x) = find (3x 4 +5x 2 +6) 7 f (x) [Example 3] (similar to Sugg Exercise 34#71) For f(x) = 3 x 2 3x + 21 Find f (x) Harder problems: [Example 4]: (similar to Sugg Exercise 34#71) For f(x) = 3 x 2 3x + 21 (A) Find the equation of the line tangent to the graph of f at x = 4 (B) Find x-coordinate of all points on the graph of f that have a horizontal tangent line [Example 5] (similar to Sugg Exercise 34#67) For f(x) = x 3 (x 7) 4 Find f (x) and find all values of x where the tangent line is horizontal Discuss common mistake Quiz 5

19 Day 24 Tue Section 34 The Chain Rule, Continued [Class Drill 12] Don t forget the easy derivative rules, part II (Section 34) Only two parts Today: Examples where the outer function is not a power function Example where the outer function is a logarithmic function [Example 1] (similar to Sugg Exercise 34#35): For f(x) = 7 ln(5x 2 30x + 65) Find f (x) Examples where the outer function is an exponential function [Example 2] Discuss the Exponential Function Rule #2 d dx e(cx) = c e (cx) Recall that we introduced it on Day 18 It could be proven at that time only using calculus techniques beyond the level of this course Example 6: Prove it again using Chain Rule [Example 3] (similar to Sugg Exercise 34#43) Chain Rule problem involving exponentials: f(x) = e ( x2 +4x 4) = e ( (x 2)2 ) (A) Find the equation of the line tangent to the graph of f at x=0 (C) Find the x-values of all points with horiz tangent line (D) Illustrate on computer (E) Discuss Bell Curve The general form of a function with a bell-shaped curve graph is: f(x) = e (polynomial) where the polynomial has degree=2 and a negative leading coefficient Day 25 X Rate of Change Class Drills [Class Drill 13a] (Exponential Function, based on Suggested Exercise 34#95) [Class Drill 13b] (Rational Function with Peak, based on Suggested Exercise 33#97) [Class Drill 13c] (Rational Function with Horizontal Asymptote, based on Sugg Exercise 33#93) [Class Drill 13d] (Square Root Function, based on Suggested Exercise 34#91) Day 26 is Exam 2 on Chapter 3 and Rate of Change Class Drills

20 Day 27 X Section 41 1 st Derivatives and Graphs Today: horizontal tangents, increasing and decreasing functions correspondence Be sure to keep f on left and f on right!! If f is positive at x = c then line tangent to graph of f at x = c tilts upward If f is negative at x = c then line tangent to graph of f at x = c tilts upward If f is zero at x = c then line tangent to graph of f at x = c is horizontal Definition of f is increasing on interval (a, b) : If a < x 1 < x 2 < b then f(x 1 ) < f(x 2 ) correspondence Be sure to keep f on left and f on right!! If f is positive on (a, b) then f increasing on (a, b) If f is negative on (a, b) then f is decreasing on (a, b) If f is zero on (a, b) then f is constant on (a, b) Discuss Reference 6 Derivative Relationships from course packet (Project on screen) Graphs of f graph of f [Class Drill 15] (similar to Suggested Exercise 41#83) Determining shape of graph of f by studying shape of graph of f (Class drill is exercise 41 #84 graph of f sign behavior of f graph of f ) Analytical example: Formula for f information about increasing/decreasing behavior of f [Example 1] (similar to Suggested Exercise 41#57) f(x) = 2x 3 3x 2 12x + 5 Find x-coordinates where graph of f has horiz tangent line Find intervals of increase & decrease information about f graph of f [Example 2] (similar to Suggested Exercise 41#61 ) function values for f & sign chart for f graph f Given information about f and f Function f is continuous for (, ) x Function values ( f(x) ) x (, 1) 1 ( 1,1) 1 (1, ) Sign chart for f ( f ) Sketch the graph of f [Example 3] (similar to Suggested Exercise 41#65) function values for f & sign info for f graph f Given information about f and f Function f is continuous for (, ) Function values f( 2) = 1, f(0) = 0, f(2) = 1 Derivative values f ( 2) = 0, f (2) = 0 Derivative sign f (x) > 0 on (, 2), ( 2,2), (2, ) Sketch the graph of f [Example 4] (similar to Suggested Exercise 41#75) f (x) graph of f graph of f Given the graph of f shown at right, (A) Find the x-coordinates of all points on the graph of x f that have horizontal tangent lines (B) Find the intervals on which f is increasing (C) Find the intervals on which f is increasing (D) Sketch a possible graph of f

21 Day 28 Tue Continuing Section 41 1 st Derivatives and Graphs Today: Local Extrema and the 1 st Derivative Test Local Extrema Define local max and local min notice that function f need not be continuous [Class Drill 16] Studying Graph Behavior Observation: If f has a local max or min at x = c, then the following three things are true: (1) f (c) = 0 or f (c) DNE (2) f(c) exists That is, the number x = c is in the domain of the function f (3) f changes sign at x = c Define words: x = c is a partition number for f mean x = c has property (1) Define words: x = c is a critical number for f mean x = c has properties (1) and (2) With terminology, we can say abbreviate our observation stated earlier: If f has local max or min at x = c, then x = c is a crit num for f and f changes sign at x = c Is the converse true? That is, is it true that If x = c is a crit num for f and f changes sign at x = c, then f has local max or min at x = c? Answer: No Consider Class Drill 15 Examples # 9,10 Both functions have x = 7 as critical number and for both functions, f changes sign at x = 7, but #9 has local max and #10 does not The problem with examples #9,#10 is that f is discontinuous at x = 7 The First Derivative Test Introduce first derivative test for local extremum at x=c If (1) f (c) = 0 or f (c) DNE (2) f(c) exists That is, the number x = c is in the domain of the function f (3) f is continuous at x = c (4) f changes sign at x = c then f has a local max or local min at x = c Notice that this test will detect all local extrema that occur at locations where f is continuous So for instance, in Class Drill 15, it would detect the local extrema in examples # 1,2,11 But the test will not detect local extrema that occur at locations where f is not continuous So for instance, in Class Drill 15, it would not detect the local max in example #9 Using the First Derivative Test [Class Drill 17] (similar to Suggested Exercises 41#17) Using the First Derivative test (sign chart for f graph of f) [Example 1] (similar to Suggested Exercises 41#41) Find partition numbers of f, critical number of f, intervals of incr/decr, and local extema f(x) = x x f (x) = 4x x = 4x(x 2 25) = 4x(x + 5)(x 5) (local max at x = 5 and x = 5 Local min at x = 0)

22 Day 29 X Continuing Section 41 1 st Derivatives and Graphs Today:More sophisticated examples involving the 1 st Derivative Test [Example 1] (Similar to Suggested Exercise 41#43) Find partition numbers of f, critical number of f, intervals of incr/decr, and local extema Function f(x) = xe ( x) with derivative f (x) = (x 1)e ( x) (f has partition number at x = 1 f has critical number at x = 1 Local max at x = 1) [Example 2] (Similar to Suggested Exercise 41#29) Find partition numbers of f, critical number of f, intervals of incr/decr, and local extema Function f(x) = 1/(x 7) 2 with derivative f (x) = 2/(x 7) 3 (f has partition number at x = 7 But f has no critical numbers! No local extrema) [Example 3] Similar to Suggested Exercise 41#85 Find partition numbers of f, critical number of f, intervals of incr/decr, and local extema Function f(x) = x + 4/x with derivative f (x) = 1 4/x 2 (f has partition number at x = 2,0,2 But f has critical numbers only at x = 2,2 Local max at x = 2, local min at x = 2) [Example 4] (Function from Class Drill 12(b), similar to Suggested Exercise 41#97) A drug is administered by pill The drug concentration (in milligrams per milliliter) in the bloodstream t hours after the pill is taken is given by the formula C(t) = 014t for 0 t 21 Day 30 t (A) find intervals of increase & decrease (B) find t-coordinate of local extrema (C) Find local extrema Round to three decimal places [Example 5] Similar to book Example, but not similar to any Suggested Exercises Skip? Find partition numbers of f and critical number of f Function f(x) = 5(x 7) 2/3 with derivative f (x) = (Critical number at x = 7 Local min at x = 7) 10 3(x 7) 1/3 X Section 42: Concavity Today: Concavity and 1 st Derivative Definition of concave up at x = c words: f is concave up at x = c meaning: The graph of f has a tangent line at x = c, and for x-values near c, the graph of f stays above the tangent line picture: Definition of concave up on an interval Definition of inflection point [Class Drill 18] Identifying Three Kinds of Graph Behavior Relationship between 1 st derivative and concavity Study Examples that show concave up f increasing, etc [Class Drill 19] Using a graph of f to get information about f Quiz 6

23 Day 31 X Continuing Section 42: Concavity Today: Concavity and 2 nd Derivatives On Day 30, we discussed that concave up on interval (a, b) f increasing on (a, b) This leads us to want to know how to figure out when f is increasing or decreasing This leads us to want to know how to figure out when the derivative of f is positive or neg This leads us to want to consider the derivative of f Introduce 2 nd derivative [Example 1] find the 2 nd derivative of the function f(x) = ln(x 2 + 6x + 13) Point out that book Example 3 considers intervals of concavity and inflection points of a function like this [Example 2] find the 2 nd derivative of the function g(x) = xe ( x) (Useful for 42#89) [Example 3] similar to sugg exercise 42#89 An ice cream company found that the price demand equation for quarts of their ice cream is p = 3e ( x) for 0 x 5, where p is the price of a quart (in dollars) and x is the number of thousands of quarts that they sell each week (A) Find the Revenue Function (B) Find the intervals of increase & decrease and the local extrema for the Revenue Function (C) Find the intervals of concavity and the inflection points for the Revenue Function Day 32 X Continuing Section 42: Concavity Today: Curve Sketching [Example 1] curving graph description of sign behavior of (f, f, f") examples: (f, f, f") = (+,, +) and (f, f, f") = (, +, ) Do a whole bunch of them [Example 2] Do exercise 42#42 function values for f and sign charts for f, f graph of f Given information about f, f, f Function f is continuous for (, ) x Function values ( f(x) ) x (, 2) 2 ( 2,2) 2 (2, ) Sign chart for f ( f ) x (, 1) 1 ( 1,2) 2 (2, ) Sign chart for f ( f ) Sketch the graph of f [Class Drill 20] Sketching a Graph of a Function from Information about its Derivatives Given the following information: f( 2) = 2 f(0) = 1 f(2) = 4 f ( 2) = 0 f (2) = 0 f (x) > 0 on ( 2,2) f (x) < 0 on (, 2) and (2, ) f"(0) = 0 f"(x) > 0 on (, 0) f"(x) < 0 on (0, ) Sketch a possible graph of f Introduce the idea of the graphing strategy [Class Drill 21] Using the Graphing Strategy to Graph a Polynomial

24 Day 33 X Section 45 Absolute Extrema discuss the Definition (page 293) discuss Theorem 2 (Locating Absolute Extrema) (page 294) discuss Theorem 1 (Extreme Value Theorem) (page 294) Introduce The Closed Interval Method (page 295) [Example 1] f(x) = x 4 6x Find extrema on [ 3,2] (Note for instructor: f (x) = 4x 3 12x = 4x(x 2 3) = 4x(x + 3)(x 3)) Started: Step 1: confirmed that interval is closed and f is continuous on that interval Step 2: Found critical values in the interval (discussed incorrect method versus correct method (using factoring) to find critical values Made list of important x-values Step 3: get list of corresponding y-values Steps 4,5: write conclusion Illustrate with computer graph [Example 2] Repeat the example with interval [-1,2] Observations: Changing the interval changes the result Simply computing y-values at all integer x-values in the interval is not sufficient Day 34 X continuing Section 45 Absolute Extrema today: What happens when extreme value theorem cannot be used? Graphical examples: do [Class Drill 22] Local and Absolute Extrema as a class Analytical examples where extreme value thm can t be used [Example 1] Same function as Day 33: f(x) = x 4 6x Find absolute extrema on (, ) (function continuous, but interval not closed) (Note for instructor: f (x) = 4x 3 12x = 4x(x 2 3) = 4x(x + 3)(x 3)) Illustrate with computer graph [Example 2] (similar to Suggested Exercises 45#51,53) Another problem where the Closed Interval Method cannot be used f(x) = 20 4x 250 x 2 Find all absolute extrema on (0, ) (interval not closed) Solution: solve using two methods: 1 st derivative information 2 nd derivative information Illustrate with computer graph (did not do this) Useful information: f(x) = 20 4x 250 x 2 f (x) = partition numbers 0,5 x3 f (x) = 1500 partition number 0 x4 Results: local max at (x, y) = (5, 10) absolute max on (0, ) is y = 10, occurs at x = 5 no absolute min on (0, ) Discuss what happens if the interval is changed to whole domain of function Quiz 7

25 Day 35 X Section 46 Optimization Today: Area and Perimeter Discuss general issues: Optimization problems are simply Max/Min problems, but they may have complications May be presented as word problems May have domains that are not closed intervals You will probably have to figure out the domain May involve more than one variable [Example 1] (Similar to Suggested Exercise 46#9,17) Find positive numbers x, y such that The sum 2x + y = 900 The product maximized (Step 1) Identify Equation I: 2x + y = 900 (Step 2) Write Equation II involving x and y and the letter P for the product Result P = xy We want to maximize P (Step 3) Solve Equation I for y in terms of x Result: New Equation I: y = 900 x (Step 4) Substitute New Equation I into Equation II and simplify to get a new equation that gives the product A as a function of just one variable x Call this function P(x) Result: P(x) = x(900 2x) (Step 5) Using Calculus, find value of x that maximizes P(x) Result: x = 225 (Step 6) Find corresponding values of y and the product Result: y = 450, product = 101,250 [Example 2] (similar to Suggested Exercise 46#34,35, using numbers from previous example) Fence problem: Farmer needs to build a fence to make a rectangular corral next to an adjacent pasture He needs only needs to fence three sides, because the fourth side has already been fenced He has 900 feet of fencing What are the dimensions of the pasture that will enclose the largest possible area? Result: x = 225, y = 450, Area is A = 101,250 square feet