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sample-midterm2-with-solutions-v2

# sample-midterm2-with-solutions-v2 - Math 113/114 SOLUTIONS...

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1 Math 113/114 SOLUTIONS TO Sample Midterm #2 Q1. 23 Find ' if cos (sin( )). Do not simplify your answer. yy x ± Solution Q2. ±² 5 3 2 4 2 Find '( ) ( ) . Do not simplify your answer. 1 xx fxi f fx x ³ ´µ ³ ·¸ Solution ± ² ± ² 4 2 24 3 4 3 3 44 32 1 2 2 1 4 2 '( ) 5 . 11 x x x x ± ² ³³¹ ³ º ³ º ³ ´ µ ¶º ´ µ ³³ · ¸ Q3. Use differentials to estimate the value of 3 7.9 Solution We know that () ( ) ' ( ) f ad x fa fad x ± ³ 12 3 33 2 3 Let ( ) . Then '( ) 3 3( ) yf x x x ± ±± ± ± ± . Since we know the value of the cube root of 8, and since 8 is close to 7.9, we let a = 8 and dx =-.1. Then substituting these values in the equation (1) above, we get 3 7.9 2 3 3 (8 .1) (8 ( .1)) (8) '(8) ( .1) 1 2 ( .1) 3( 8 ) 2( ) 34 10 1 2 120 119 1 120 119 7.9 1 120 f f ff ±³ ² ´³ µ ² µ² µ ±³ µ² µ ± ¶´ ] 3 ) cos( )) sin(sin( ))[ cos(sin( 2 2 3 3 3 x x x x y ± ± ² ³ ´

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2 Q4. Find the second derivative of ±² 2 1 () 1 fx x ± ² in the simplest form. Solution Q5. Determine the following limits: 5 42 0 cot(2 ) 3 50 a. lim . b. lim csc( ) 6 2 10 x xx ± ± ± ±± ² ²± Solution a. 0 cot(2 ) lim csc( ) ± ± ± ± = 0 cos(2 ) sin( ) lim sin(2 ) 1 ± ± ± ± = 0 cos(2 ) sin( ) 1 lim 11 s i n ( 2 ) ± ± ± = 0 cos(2 ) sin( ) 2 1 lim 1s i n ( 2 ) 2 ± ±±± ± ± = 00 0 1c o s ( 2 ) s i n ( ) 2 lim lim lim 21 s i n ( 2 ) ± ± = 111 . 22 ±±± ² b. 2 3 2 2 3 2 2 1 2 ) 1 ( 2 ) 1 ( 2 1 ) ( ) 1 ( ) ( ± ± ± ± ² ³ ´ ² ± ² ³ ´ µ ± ´ x x x x x f x x f 5 2 2 2 5 2 2 2 2 2 5 2 2 5 2 2 2 3 2 2 5 2 2 3 2 ) 1 ( 1 2 ) 1 ( 1 2 ) 3 1 ( ) 1 ( ) 1 ( 3 ) 1 ( ) 2 ( ) 1 ( 2 3 ) 1 ( 1 ) ( x x x x x x x x x x x x x x x f ± ² ³ ± ² ³ ² ± ´ ± ² ³ µ µ · ¸ ¸ ¹ º ± ´ ² ± ² ³ µ µ · ¸ ¸
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sample-midterm2-with-solutions-v2 - Math 113/114 SOLUTIONS...

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