Lincoln County Schools Patriot Day Instructional Expectations Patriot Day 5 School: LCHS Course/Subject: AP Calculus Teacher: Susan Harris

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1 Lincoln County Schools Patriot Day Instructional Expectations Patriot Day 5 School: LCHS Course/Subject: AP Calculus Teacher: Susan Harris Learning Target: 4..1 I can use implicit differentiation to find the derivative of equations that may not be functions 4.. I can use implicit differentiation to find slopes of curves and apply them to write equations of tangent and normal lines 4..3 I can apply derivatives found by implicit differentiation 4..4 I can use implicit differentiation to find nd or higher order derivatives I can set up and solve a problem to optimize values I can use related rates to find values in terms of time. Lesson Expectations/Standard: College Board Curriculum: Concept of the derivative Derivative presented graphically, numerically, and analytically. Derivative interpreted as an instantaneous rate of change. Derivative defined as the limit of the difference quotient. Relationship between differentiability and continuity. Derivative at a point Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. Tangent line to a curve at a point and local linear approximation. Instantaneous rate of change as the limit of average rate of change. Approximate rate of change from graphs and tables of values. Derivative as a function Corresponding characteristics of graphs of ƒ and ƒ. Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ. The Mean Value Theorem and its geometric interpretation. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Second derivatives Corresponding characteristics of the graphs of ƒ, ƒ, and ƒ. Relationship between the concavity of ƒ and the sign of ƒ. Points of inflection as places where concavity changes. Computation of derivatives Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Derivative rules for sums, products, and quotients of functions. Links to Other Assignment Options (websites or programs): Patriot Day 5 lesson Links to Resources and Support: Desmos graphing calculator: * Android Devices: * Apple Devices: Teacher Support: (list s &/or phone numbers here): susan.harris@lincoln.kyschools.us Phone: *Reminder: Assignments are due back to teachers the next day we are in school, if possible, or within 1 week of the Patriot Day.

2 Calculus r Am0g1q6a RKEuNtbah ysaoufytzwtarroeh ELKL]Cs.y c taulela ercixgthathsh vr_emsreyryvceddm. Patriot Day 5 Solve each optimization problem. Name 1) A farmer wants to construct a rectangular pigpen using 300 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area? A) 80 ft (perpendicular to wall) by 140 ft (parallel to wall) B) 77 ft (perpendicular to wall) by 146 ft (parallel to wall) C) 75 ft (perpendicular to wall) by 150 ft (parallel to wall) D) 76 ft (perpendicular to wall) by 148 ft (parallel to wall) ) A company has started selling a new type of smartphone at the price of $ x where x is the number of smartphones manufactured per day. The parts for each smartphone cost $40 and the labor and overhead for running the plant cost $6000 per day. How many smartphones should the company manufacture and sell per day to maximize profit? A) 1300 B) 800 C) 750 D) 550 3) A geometry student wants to draw a rectangle inscribed in a semicircle of radius 7. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the student can draw? A) 48 B) 50 C) 47 D) 49 4) A rancher wants to construct two identical rectangular corrals using 100 ft of fencing. The rancher decides to build them adjacent to each other, so they share fencing on one side. What dimensions should the rancher use to construct each corral so that together, they will enclose the largest possible area? A) 35 B) 9 C) 31 D) 5 ft (non-adjacent sides) by 10 ft (adjacent sides) ft (non-adjacent sides) by 14 ft (adjacent sides) ft (non-adjacent sides) by 38 3 ft (non-adjacent sides) by 50 3 ft (adjacent sides) ft (adjacent sides) C Kj0V1R6A \KYuot\aQ VSLobfItNwaaurQej xlbllcm.b n LAYljlW JrKiUgrh\t`sW UraefsseZrlvzemdZ.` T vmqafdeew YwRiot[hc hiunmfmipnuintnel cc_atloc]u^lluhsx. -1-

3 5) Engineers are designing a box-shaped aquarium with a square bottom and an open top. The aquarium must hold 916 ft³ of water. What dimensions should they use to create an acceptable aquarium with the least amount of glass? A) 1 ft by 1 ft by C) 3 ft by 3 ft by ft tall B) 0 ft by 0 ft by ft tall 100 ft tall D) 18 ft by 18 ft by 9 ft tall 6) A cryptography expert is deciphering a computer code. To do this, the expert needs to minimize the product of a positive rational number and a negative rational number, given that the positive number is exactly 6 greater than the negative number. What final product is the expert looking for? A) 0 B) 7 C) -8 D) -9 7) An architect is designing a composite window by attaching a semicircular window on top of a rectangular window, so the diameter of the top window is equal to and aligned with the width of the bottom window. If the architect wants the perimeter of the composite window to be 1 ft, what dimensions should the bottom window be in order to create the composite window with the largest area? 17 A) 4 + p ft (width) by p ft (height) B) p ft (width) by 1 ft (height) 4 + p 16 C) 4 + p ft (width) by p ft (height) D) p ft (width) by 6 ft (height) 4 + p 8) A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume? A) 5 in B) 6 in C) 3 in D) 8 in 9) A geometry student wants to draw a rectangle inscribed in the ellipse x + 4y = 36. What is the area of the largest rectangle that the student can draw? A) 39 B) 37 C) 38 D) 36 U LC0B1I6t EKduwtBaB RSHo[fytOwwaQrZeQ zlfljcp.n Q ZAVlVld JrjiSgOhUtQsM nrue\sxewrnvlefdn.n D wmfaedwek gwgirtrhq miznqfdivnxictfec ECianlDciuvlpuLs^. --

4 10) Two vertical poles, one 8 ft high and the other 16 ft high, stand 45 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope? A) 17 ft from the short pole (or 8 ft from the long pole) B) 15 ft from the short pole (or 30 ft from the long pole) C) 18 ft from the short pole (or 7 ft from the long pole) D) 19 ft from the short pole (or 6 ft from the long pole) For each problem, use implicit differentiation to find dy in terms of x and y. dx 11) 4x - 3y = 1 dx = 4x 3y dx = 3y 4x dx = -8x dx = 4x - 3y 1) 3y + 3 = 3x dx = 1 y dx = 1 x dx = y dx = y + 1 x 13) 1 = 5x + 3y 3 dx = 10x 5x + 3y 3 dx = - 9y 10x dx = 1 5x + 3y 3 dx = - 10x 9y 14) 4x 3 - y = 4 dx = 3x y dx = x3 - y dx = y 3x dx = -3x 15) y + 3 = 4x dx = 4x y dx = y 4x dx = x dx = y + 3 4x 16) -y + 5 = 5x 3 dx = -y + 5 5x 3 dx = - 4y 15x dx = 3 x dx = - 15x 4y L EB0W1o6G gkhujt]ah OSfoefatIw_a_r_eW NLgLFC_.r b darlclj qrbixguhvtfse NrkeesreprSvieqds.M _ `MxaEdxeZ tw_iftmhi QITn`fkiDnGiBt`eF IC]aflsckuWlyuEsm. -3-

5 17) 4y + 1 = 4x dx = y x dx = x dx = 4y + 1 4x dx = x y 18) -y = 3x 3 dx = - 3x y dx = -y x 3 dx = - y 3x dx = 3 x For each problem, use implicit differentiation to find d y in terms of x and y. dx 19) 3x 3 = -y + 3 A) d y dx = 36x y 4-108x y + 81x + 36x 6 y -8y y 4-54y + 7 B) d y dx = -7xy - 81x 4 16y 3 C) d y 16y + 7yx = dx 81x 4 D) d y dx = -7xy4 + 16xy - 16x + 34x 4 y -8y y 4-54y + 7 0) = x + 5y A) d y dx = -5y - x 5y 3 B) d y dx = 30y x C) d y dx = D) d y dx = -x y 4-40x y x x 4 y + 75x y y 6-4x 3-0xy - 40y x x 4 y + 75x y y 6 1) 5 = 3x - 3y A) d y dx = - 1 4y 3 B) d y dx = C) d y dx = D) d y dx = 4y -15x + 15y + 50y 9x 3-7x y + 7xy 4-9y 6 -x + y + y x 3-3x y + 3xy 4 - y 6 ) 3x 3 + 3y = 3 A) d y dx = -1xy - 9x 4 4y 3 B) d y 4y + 1yx = dx 9x 4 C) d y dx = 3x + yx 3 + y 3 D) d y dx = -6x a vm0l1v6c FK]uHtFa` usoowf^tqwza_rbe` llhlecx.q s uatlclk irxiqgyhvtksy RrWeBsfe[rAvaeydk.Q s kmsazdkej bwxittohg nimnufti_npict\e^ NCfaBlVc]uSlWuAsN. -4-