University:
Stanford University
Course:
MATH 19 | Calculus
Academic year:

2023
Views:

11

Pages:

3
Author:

MathPaladin

Questions and Answers Sheet 95 Fractions Question #1 How to integrate  (e ex x − 2 )( e2 x + 1) dx using partial fractions? Answer: Substitution with e = y then partial fractions x x = ln y , dx =  1 dy y 1 ( y − 2 ) ( y 2 + 1) dy Search A, B , C so that 1 ( y − 2) ( y 2 + 1) = A By + C + 2 y − 2 y +1 A + B ) y 2 + ( C − 2B ) y + A − 2C ( = ( y − 2) ( y 2 + 1) ( y − 2) ( y 2 + 1) 1 So A = − B , C = 2B and A − 2C = −B − 4B = 1 1 1 2 Then B = − , A = , C = − 5 5 5 1  dy = ( y − 2) y2 + 1 ( )  1 1 1 2y 1 =  dy −  2 dy − 2  2 dy  = 5 y−2 2 y +1 y +1  1 1  =  ln y − 2 − ln y 2 + 1 − 2arctan y  + c 5 2  x Then substitute y with e Question #2 Simplify the fraction 35 50 Answer: Fractional simplification. We have that, Consider 35 50 35 50  5Ч 7 7  5Ч 10 10 Thus, 35 7 . 50 10 Question #3 If y = x + 2 x and 3 dx dy = 5 , how do you find when x = 2 ? dt dt Answer: By Implicit Differentiation, dy = 70 dt With respect to t, we can differentiate, ( ) ( ) dy d 3 dx = x + 2 x = 3x 2 + 2 dt dt dt dx =5, by plugging in x = 2 and dt 2 = 3 ( 2 ) + 2  ( 5 ) = 70   Question #4 What is 0.04% into a fraction? Answer: 𝑎% = 𝑎 100 0.04% = ∴ 0.04 100 0.04 100 4 1 × = ⇒= 100 100 10000 2500 Question #5 Divide the sum of 8 65 and by their difference. 3 12 Answer: 8 65 8 65 + 32 97 65 65 + = = and = . Difference of and 3 12 3 12 12 12 12 8 65 8 65 − 32 33 97 33 97 12 97 = − = =   =  = 3 12 3 12 12 12 12 12 33 33 Sum of Question #6 What fraction of a month (30 days) is a week? Answer: Step 1 We know that 1 month=30 days 1 week=7 days Step 2 Hence, the required fraction is 7/30 Question #7 135 % as a fraction is ____. 11 ; 20 7 B )1 ; 20 19 C )1 ; 20 D )1 A)1 Answer: The correct option is B 17 7 135 27 135% = = (simplifying by 5) = 1 20 20 100 20