PreCalc: simple limits and differentiation?

Question

I just wanted to clarify a couple of questions; I don't need explanations unless I'm incorrect!

Algebraically determine:
(a) lim x-->2 (x^2) - 3x +4 Ans: 2?
(b) lim x-->0 ((3x-2)^2) - 5x + 8 Ans: 12?
(c) lim x-->5 ((x^2) - 3x - 10) / (x - 5) Ans: 7?

Can someone just simply explain how to do calculate the first derivate, maybe use the following as an example?
(d) f(x) = 1 / (x^2)

TIA!! Love Tessa xxx

Woot!! Thanks to both of you. Can't i simplify the last question any further than what you have provided?? (First to answer gets best ans points!)

Thanks!! Love Tessa xx

Answer 1

a) correct
b) correct
c) correct

d)

f(x) = x^-2
f '(x) = -2x^-3
f '(x) = -2/x^3

Answer 2

I guess I don't need to explain a-c since the others have said that they are correct.

To explain part (d) to you, let me try and explain it to you

1/(x^2) can be rewritten as x^(-2), since the power will be negative when it is being moved from the denominator to numerator, and vice versa.

To perform the differentiation, first place the power of x as a constant in from of your expression. This makes the first portion of your answer to be -2 * x^(-2)

Then, take the power of x, and subtract it by 1. So, the next portion of your answer will be -2 * x^(-3)

To change it beck to the form that u started with (having a positive power of x), just place x at the numerator and remove the negative power on x. This gives the final answer

-2/(x^3)

Hope this helps

Answer 3

All those answers for the limits are correct.

to calculate the first derivative,the formula is:-

lim(h-->0) (f(x+h)-f(x)) / h

so,using the first derivative,
f(x)=1/(x^2)

lim(h--->0) (1/((x+h)^2)-1/(x^2)) / h

lim(h-->0) (x^2-x^2-2xh-h^2) / hx^2(x+h)^2

lim(h-->0) -h(2x+h) / hx^2(x+h)^2

lim(h-->0) -(2x+h) / x^2(x+h)^2

as lim(h-->0),so it will become

-2x / x^4= -2 / x^3

so,the final solution will be -2 / x^3.

hope it will help u!

Answer 4

Well Done!

d) IF Y=U^M THEN Y'=M * U^(M-1)

f(x)=x^-2

f'(x)=-2/x

Answer 5

(a)~(c) correct. well done!

(d) f(x) = 1/(x^2)
= x^(-2)
=>f'(x) = -2(x^(-3))