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Mathematics and Statistics
MATH 1013
Peter Taylor

MATH 1013 Applied Calculus Section D Fall 2012 Test 2/ November 5, 2012 IP-No.: Student Name: You have 50 minutes to solve the following problems: Show your complete work. The maximal mark on this test, including any bonus points, is 100. Permitted aid: helpsheet of standard letter size with notes and formulas. NO CALCULATOR (graphical or nongraphical) is allowed on this test. Problem 1) (24 marks) Differentiate each function with respect to the independent variable: ..(y)::: 2 .s.ecC...r) ~ I( x).::: d) f CX) .. --{';"X~-I .S~f -4 6 f)fcx) :::: .e>\s; Vl( x--) e) L,cs.).::. s +-2 r ~(5 ".f-'0). 2 s·Cs'-'2)-1·(>2"-b) f~x):: L,(!):: > ::.s~V?,ex) f~X) L =O)~(..t) L/(X):::.2 X.( 1-l- )(") ~~ -e ..J-K ·-e You are given a formula for the derivative f '(x):ks) I fcx) = 3x+-t Use the formula to a) locate any/all critical points o=0).i.e. points x with f '(x) b) determine the intervals over which f is decreasing, and c) sketch a graph off based on your answers from a) and b). r a.) (ex) :.-0 =-> X;:: - k3 S) ~~ cv '-C~De-. ,·5 v 0. Then the object whose position is y(t) is ... i) speeding up ii) slowing down iii) moving toward the origin iv) moving away from the origin v) none of the above 1 b) Suppose y = sin- (x). Then ... r -2 I i) 'f =- S ,•"'? (X) ii) 7 ==- cscc.x) ·co fcx) iii) '7 I= ' I iv) / - -.c;--=={-)(--?_l v) none of the above 1 c) Suppose (xo, yo)is on the graph of the invertible function f. Then =f" )'(yo) -2 I i) - f C'70 ) · F (Ko) ( iii) iv) v) none of the above . Then ... d) Suppose, 7 .:: (VI( 3x ¥- I) f j_ ~ i) ii) 7 ~ 3 .X o~I - I iii) iv) .::: 3 I = 7' IV!(]x-+1) v) none of the above. Problem 4) (20 marks). Use implicit differentiation to show that the x-intercept andy-intercept of any tangent line to the curve 1--J7 ::-fC' sum up to c, where c is any positive constant. .0:_(-Ck T--C7{k ) ) = d o(..x dx(-(2) / _}_ I ~ +- 7 c{'1 ::0: 2-fX' 2 -iyrx) o
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