DIFFERENTIAL CALCULUS PROBLEMS WITH SOLUTIONS- LIMITS

DIFFERENTIAL & INTEGRAL CALCULUS
by
FELICIANO AND UY

STEP BY STEP SOLUTIONS


CHAPTER 1 | LIMITS



EXERCISE 1.1

1. If f ( x )=x −4 x , find (a) f (−5 ) (b) f ( y +1 ) (c) f ( x + Δx )
2 2



(d) f ( x +1 )−f ¿)


Solution:

(a) f (−5 ) (b) f ( y 2+1 )

f ( x )=x 2−4 x f ( x )=x 2−4 x
2
f (−5 )=(5)2 −4 (−5 ) f ( y +1 ) =( y +1 ) −4 ( y +1 )
2 2 2



f (−5 )=25+20 f ( y 2+1 ) = y 4 +2 y 2 +1−4 y 2−4

f (−5 )=45 f ( y 2+1 ) = y 4 −2 y 2 −3



(c) f ( x + Δx )

f ( x )=x 2−4 x

f ( x + Δx )=¿ ( x + Δx )2−4 ( x + Δx )

f ( x + Δx )=( x+ Δx ) [ ( x+ Δx )−4 ]

, (d) f ( x +1 )−f ¿)
2
f ( x )=x −4 x

f ( x +1 )−f ( x−1 )=¿ [( x +1 )2−4 ( x+1 ) ¿−[ ( x−1 )2−4 ( x−1 ) ]

f ( x +1 )−f ( x−1 )=¿ ( x 2+ 2 x +1−4 x−4 ¿−(x 2−2 x+1−4 x + 4)
2 2
f ( x +1 )−f ( x−1 )=x −x +2 x +2 x+1−1−4 x+ 4 x−4−4

f ( x +1 )−f ( x−1 )=x 2−x 2 +2 x +2 x+1−1−4 x+ 4 x−4−4

f ( x +1 )−f ( x−1 )=4 x−8

f ( x +1 )−f ( x−1 )=4(x −2)




x2 +3
2. If ¿
x
, find x as function of y . 3. If y=tan ⁡¿π), find x as a function of y .

Solution: Solution:
2
x +3
y= y=tan ⁡¿)
x

x y =x +3
2
x +π ¿ tan−1 y
2
x −xy +3=0 x ¿ tan−1 y −π

a=1 , b=− y ∧c=3

−b ± √ b −4 ac
2
x= `
2a

−(− y ) ± √(− y ) −4 ( 1 ) ( 3 )
2
x=
2( 1)

y ± √ y −12
2
x=
2