Assignment 6.2. Calculus F a2l0i1]6^ LKduFtHaz esnopfctqwzatreet ELOLeCq.h ] gaplhlw BrJiKg\hTtjsJ brmehsdeurpvreqdu. Name ID: 1.

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1 Calculus F al0i1]6^ LKduFtHaz esnopfctqwzatreet ELOLeCq.h ] gaplhlw BrJiKg\hTtjsJ brmehsdeurpvreqdu. Assignment 6. Solve each optimization problem. Name ID: 1 Date Period 1) 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 $0 and the labor and overhead for running the plant cost $000 per day. How many smartphones should the company manufacture and sell per day to maximize profit? ) 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 5 greater than the negative number. What final product is the expert looking for? 3) Which points on the graph of y = - x are closest to the point (0, 3)? ) Which point on the graph of y = x is closest to the point (6, 0)? j _D0L1E6\ SKQuPtUaJ esropf\tcwxarrtet GLfLaCb.f X CA[lul[ hrvitgwhwtzso mr[edsue^r`vyeidp._ ^ EMZaed]eP bwviat\hl ai\n\fbidnziwtaea xciamllcrupl[uesp. -1-

2 5) Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. 6) There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees? 7) Maximize the product xy, subject to the constraint that x+y=0 For each problem, use implicit differentiation to find dy at the given point. 8) = x + y at (-1, 1) 9) = x 3-3y 3 at (1, -1) 10) x - y = at (-1, 1) 11) 5x + y 3 = 3 at (1, -1) 1) -5xy + 3y = x at (1, -) 13) 3y 3 = x - x 3 y 3 at (-1, 1) 1) xy = 3x + y 3 at (1, -1) k fl0o1w6m PK[uFtpaC CSfoUfGtrw`a\rve[ HLwLHCI.U ` JAPlmly YrnixgghVtesz urrejs]ecrvvkebd].` b DMGaJdBea fwhittzhk miwntf_ijnoiptiea kckamllcyutlpuvsx. --

3 For each problem, find the open intervals where the function is concave up and concave down. 15) f (x) = x - 8x ) f (x) = x - 3x + 17) y = -x + x + 18) f (x) = x + x 3-3x - find the velocity function v(t) and the acceleration function a(t). 19) s(t) = -t 3 + 8t find the times t when the particle changes directions. 0) s(t) = t - 6t c ez0c1k6g OKduptfaz qswomfdtlwvaxr\eh RLOLHCy.f F ZAHlill qrliygnhdtqsl yroepsherruvpeedd.u Z zmoaydres owciotphr jiqnvfniqn^iytvee QC`anlzcBudlGuksk. -3-

4 find the intervals of time when the particle is moving down and moving up. 1) s(t) = -t + 11t 3 find the times t when the acceleration is 0. ) s(t) = t - 1t 3 find the intervals of time when the particle is slowing down and speeding up. 3) s(t) = t - 15t 3 find the position, velocity, speed, and acceleration at the given value for t. ) s(t) = -t 3 + 3t - 10t; at t = 6 find the maximum speed and times t when this speed occurs over the given interval. 5) s(t) = -t 3 + 0t - 100t; 3 t 9 f nm0c1v6y VKAuxtJap bswomfdtiw`agrvep nlflqcz.g E TA`lilU greifg[h_twsa `rjessjegryvyecdz.t N CMNaZdheG WwCiwtnhx iivnjf`i]nzi`tkep bcuaklacauxlbu`sl. --

5 Evaluate each limit. 6) lim x - x x - x 3 7) lim x - x f(x) 8 f(x) x x 8) lim x x - 6 x - 1x ) lim x 1 (-x - ) Use the definition of the derivative to find the derivative of each function with respect to x. 1 30) y = - x - 3 m VZ0L1o6u ykguutwao PSroEfYtlwEakr^eL VLlLKCS.k G HAUl]lr lraing^hftdsx FrCeAsFeqr[vIerde.P h smqatdbes Lw^iptYhl ui_ncfzixnai\tqet rcjapl_ckuxlougsa. -5-

6 Answers to Assignment 6. (ID: 1) 1) p = the profit per day x = the number of items manufactured per day Function to maximize: p = x( x) - (0x + 000) where 0 x < Optimal number of smartphones to manufacture per day: 350 ) P = the product of the two numbers x = the positive number Function to minimize: P = x(x - 5) where - < x < Smallest product of the two numbers: - 5 3) d = the distance from point (0, 3) to a point on the parabola x = the x-coord. of a point on the parabola Function to minimize: d = x + ( - x - 3) where - < x < Points on the parabola that are closest to the point (0, 3): ( -, 7 ) (,, 7 ) ) d = the distance from point (6, 0) to a point on the curve x = the x-coordinate of a point on the curve Function to minimize: d = (x - 6) + ( x) where - < x < ( Point on the curve that is closest to the point (6, 0): 11, ) 5) 3, 6 6) 15 7) 9) dy 13) dy y = -1 x = -1 y = 1 = 1 3 = ) dy 1) dy x = -1 y = 1 y = -1 = - 11) dy = ) Concave up: (-, ) Concave down: No intervals exist. y = -1 16) Concave up: ( -, - ), (, ) Concave down: ( -, ( 17) Concave up: - 3 3, 3 ) ( 3 Concave down: -, - 3 ) ( 3, 3 18) Concave up: (-, -1), ( 1 ) (, Concave down: -1, 1 ) = ) ) 3, 8) dy 1) dy 19) v(t) = -3t + 16t, a(t) = -6t ) Changes direction at: t = {13} 1) Moving down: t > ) Acceleration zero at: t = {0, 6}, Moving up: 0 < t < 3) Slowing down: 15 < t < 5 15, Speeding up: 0 < t <, t > 5 ) s(6) = -108, v(6) = 8, speed at 6 = 8, a(6) = 10 5) Maximum speed: { at t = 0 3 } 8) - 9) ) 7) - 30) dy = x - 1x + 9 x = -1 y = 1 y = - = - 1 = 3 w QU0F1q6A hkxultoat [SwovfxtPw]aZr\el [LALZC\.b V ia_lmlx Irvi\gShLtbsT OrzeKsHejrQvZeydp.w A AMiaGdlev `wiiutfhv VIEn^fai[njizt\en `CYailgcQuHlpuKsm. -6-