1007_Test2_Sol

1007_Test2_Sol - MATH 1007 A Test #2 October 20, 2006...

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Unformatted text preview: MATH 1007 A Test #2 October 20, 2006 Solutions Please show all work for all questions! Part A Short Answer [22 marks] 1. [3 marks] A table of trigonometric functions and their derivatives is below. Please fill in 2 of the blanks. Function Derivative sin x cos x cos x- sin x tan x sec 2 x csc x- csc x cot x sec x sec x tan x cot x- csc 2 x 1 2. [9 marks] Calculate the first derivative of each of the following functions: (a) [2 marks] f ( x ) = 3 1 + tan x f ( x ) = 1 3 (1 + tan x )- 2 / 3 (sec 2 x ) (b) [2 marks] g ( x ) = sin (( x + 1) 2 ) g ( x ) = cos ( ( x + 1) 2 ) 2( x + 1)(1) (c) [3 marks] y = (2 x- 5) 4 (8 x 2- 5)- 3 Solution 1 (Product Rule) y = (2 x- 5) 4 (- 3)(8 x 2- 5)- 4 (16 x ) + (8 x 2- 5)- 3 4(2 x- 5) 3 (2) Solution 2 (Logarithmic Differentiation) y = (2 x- 5) 4 (8 x 2- 5)- 3 ln y = ln (2 x- 5) 4 (8 x 2- 5)- 3 = ln (2 x- 5) 4 + ln (8 x 2- 5)- 3 = 4 ln(2 x- 5)- 3 ln(8 x 2- 5) 1 y y = 4 1 2 x- 5 (2)- 3 1 8 x 2- 5 (16 x ) y = y 8 2 x- 5- 48 x 8 x 2- 5 y = (2 x- 5) 4 (8 x 2- 5)- 3 8 2 x- 5- 48 x 8 x 2- 5 (d) [2 marks] y = x x y = x x ln y = ln( x x ) = x ln x 1 y y = x 1 x + ln x (1) y = y [1 + ln x ] y = x x [1 + ln x ]...
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1007_Test2_Sol - MATH 1007 A Test #2 October 20, 2006...

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