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NYC College of Technology, CUNY Mathematics Department Spring 2023 MAT 1475 Final Exam Review Problems Revised by Prof. Kostadinov, Prof Africk and Prof Colucci: Fall 2010 through Fall 2015 Revised by Prof. Africk , ColucciDouglas, Johnstone, limiteriblyumr Taraporevala: Spring 2023 Evaluate the following Limits: x . e 1 2 x 7x_5x2 _ 2x1+Cos4x 4 x 6X23 aLm b' _]_ ch d-[_]_ )xl>0 Sin3x )l-flsn}[[3- + X3X2 +6 )x-Ia 1_e4x )li'in-l 5 +6X+2X2 +5 #3 Find the derivatives of the following functions using the Limit definition of derivative: a) _/'(x) = 2.1:2 Sr b) for) = 2x2 + 3x 4 Find the derivative y\" = E of the following functions, using the derivative rules: (it a)y=Cas"6x b)y=x-Sin4x c)y=x-C053x d)y=x-ez't e)y=x-e"' f)y=5in'1[x2] g)y=Tan1{x3] E Find the derivatives of the following functions: a) Ill-0216369617 b) 1%}: lanx} C) fix): :rz ? d) W) 3 e3\" c05(5x) elflxl: 22X_1( x _ x 1 E Find the derivative _l-" = 1: of the following functions: I'. . Cl.) _1-' : Ir b) J: : {Sinxyh C) Alf: (r-u.\" #_Q Find the equation of the tangent line, in slope-intercept form, to the following curves: a) fix): 2.r'+ 5.x: +6 at {4.9) b) fix}: 4,\\',\\': at {1.3} #_Z Using implicit differentiation, find the equation of the tangent line to the given curve at the given point: a) 3ft"? 3y I? = 5x +14 at {l,3} b) y: 7..ty+.1rj 2..t = 9 at {0.3) Q (a) Find the equation of the tangent line to y = v'x + 3 at x = 6 (ID) Find the differential dy at y = w: + 3 and evaluate it for x = 6 and dx = 0.3 Wire of length 20m is divided into two pieces and the pieces are bent into a square and a circle. How should this be done in order to minimize the sum of their areas? Round your answer to the nearest hundredth. 3t 10 Find the dimensions of a closed box having a square base with surface area 12 and maximal volume 31: 11 If a snowball melts so its surface area decreases at a rate of 1cm2/min , find the rate at which the diameter decreases when the diameter is 6 cm. 31: 12 The radius of a sphere increases at a rate of 3 in/Sec. How fast is the volume increasing when the diameter is 24in? 1t 13 The radius of a cone is increasing at a rate of 3 inches/sec, and the height of the cone is 3 times the radius. Find the rate of change for the volume of that cone when the radius is 7 inches. #14 Sand pours from a chute and forms a conical pile whoSe height is always equal to its base diameter. The height of the pile increases at a rate of 5 feet/hour. Find the rate of change of the volume of the sand in the conical pile, when the height of the pile is 4 feet