For the functions given below, find and simplify completely: g(x+h)-g(x)/h (here, h is some parameter)

Mathematics

1 answer:

maksim [4K]2 years ago

4 0

Answer:

a. 2

b. 4x -1 +2h

Step-by-step explanation:

We can find the difference quotient for the given cases by first solving for the difference quotient in the general case. That solution can then be applied to the given cases.

<h3>general solution</h3>

For f(x) = ax² +bx +c, the difference quotient is ...

$\dfrac{f(x+h)-f(x)}{h}=\dfrac{(a(x+h)^2+b(x+h)+c)-(ax^2+bx+c)}{h}\\\\=\dfrac{(ax^2+2axh+ah^2+bx+bh+c)-(ax^2+bx+c)}{h}=\dfrac{2axh+ah^2+bh}{h}\\\\=\underline{2ax+b+ah}$

For g(x) = 2x+6, we have a=0, b=2, c=6. Substituting these values into the above formula, we find ...

(g(x+h) -g(x))/h = 2·0·x² +2 +0·h

(g(x+h) -g(x))/h = 2

For g(x) = 2x² -x, we have a=2, b=-1, c=0. Substituting these values into the above formula, we find ...

(g(x+h) -g(x))/h = 2·2·x +(-1) +2·h

(g(x+h) -g(x))/h = 4x -1 +2h

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Po is trying to solve the following equation by completing the square: 49x^2+56x-64 = 0. He successfully rewrites the above equa

ohaa [14]

Answer:

The answer is 91

Step-by-step explanation:

We have the following equality:

$49x^2+56x-64=a^2x^2+2abx+b^2-c$

Then a must satisfy that $49=a^2$ . So,

$a=7$ or $a=-7$ .

1) If a=7, then, it follows for the first equality that 56=14b. Then b=4. Finally, substituting b=4 in the first equality we obtain that -64=16-c. So, c= 80. We conclude that

$a+b+c=7+4+80=91$ .