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Evaluate the function at the indicated values.

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2 Answers

1)
a.  3(x2)2 - 4  = 3x4 - 4
b.  3(3x2 - 4)2 - 4 = 3(9x4 - 24x2 + 16) - 4 = 27x4 - 72x2 + 48 - 4 = 27x4 - 72x2 + 44
c.  3(x-h)2 - 4 = 3(x2 - 2xh + h2) - 4 = 3x2 - 6xh + 3h2 - 4
d.  (3x2 - 4) - (3h2 - 4) = 3x2 - 3h2 -4 + 4 = 3x2 - 3h2 = 3(x2-h2) = 3(x+h)(x-h)
e.  (g(x+h) - g(x)) / h = (3(x+h)2 - 4 - (3x2-4))  / h  = (3(x2 +2xh + h2) - 4 -3x2 + 4)) / h
 
(3x2 + 6xh + 3h2 -3x2) / h  = (6xh + 3h2)/h  = 6x + 3h
 
The limit as h->0 of 6x + 3h is 6x, the first derivative of g(x) = 3x2 - 4.
 
 
The other problem is similar.

Comments

Good Day sir,
 
Is it okay if my answer for letter d is 3x2 - 3h2 instead of 3(x+h)(x-h)?
1)a) g(x2) = 3(x2)2 - 4 = 3x4 - 4
b) g(3x2-4) = 3(3x2 - 4)2 - 4 = 3(9x4 - 24 x2 + 16) - 4= 27x4 - 72x + 48- 4= 27x4 - 72x+44
C)g(x-h)= 3(x-h)2 - 4 = 3(x2 - 2hx + h2) -4 = 3x2 -6hx + 3h2 -4
d)g(x)-g(h)= 3x2-4-3h2+4=3x2 - 3h2
e)(g(x+h)-g(x))/h=((3(x+h)2-4-3x2+4)/h=(6xh+3h2)/h=6x+3h
 
2)4-x should be greater or equal to 0 , then x is less or equal to 4
a) G(4-x)= √(4-4+x)=√(x) where x ≥ 0
b) G(4-x2)=√(4-(4-x2)=√(x2)= x if and only if x ≥ 0, -x when x ≤ 0 as the square root should be always positive
 
 c) G(4-x4)= √ (4-4+x4)=x2
d) G(4x-x2)= √(4-4x+x2)=√(x-2)2=x-2 where x ≥ 2 and -x+2 when x ≤ 2
e) G ( -x4-4x2)= √( 4+x^4+4x2)= √(x2 + 2)2 = x2 + 2`

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