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2 years ago

14

Consider the function f left parenthesis x right parenthesis equals 3 x squared minus 5 x minus 2f(x)=3x2−5x−2 and complete part

s (a) through (c). (a) find f left parenthesis a plus h right parenthesisf(a+h); (b) find startfraction f left parenthesis a plus h right parenthesis minus f left parenthesis a right parenthesis over h endfraction f(a+h)−f(a) h; (c) find the instantaneous rate of change of f when aequals=44. (a) f left parenthesis a plus h right parenthesisf(a+h)equals= nothing (simplify your answer. do not factor.)

1 answer:

GuDViN [60]2 years ago

8 0

Given f(x)=3x²-5x-2

a) To find f(a+h) replace x with a+h in the given function. So,

f(a+h)=3(a+h)²-5(a+h)-2

=3(a²+2ah+h²)-5(a+h)-2 By using the formula (x+y)²=x²+2xy+y².

=3a²+6ah+3h²-5a-5h-2 By distributing property.

b) Similarly to find f(a) we need to replace x with a. So,

f(a)=3a²-5a-2

So, f(a+h)-f(h)= (3a²+6ah+3h²-5a-5h-2)-(3a²-5a-2)

=3a²+6ah+3h²-5a-5h-2-3a²+5a+2.

=6ah+3h^2-5h (All other terms has been cancel out)

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Complete the table of values

Agata [3.3K]

Both problems give you a function in the second column and the x-values. To find out the values of a through f, you need to plug in those x-values into the function and simplify!

You need to know three exponent rules to simplify these expressions:
1) The negative exponent rule says that when a base has a negative exponent, flip the base onto the other side of the fraction to make it into a positive exponent. For example, $3^{-2} =
\frac{1}{3^{2} }$ .
2) Raising a fraction to a power is the same as separately raising the numerator and denominator to that power. For example, $(\frac{3}{4}) ^{3} = \frac{ 3^{3} }{4^{3} }$ .
3) The zero exponent rule<span> says that any number raised to zero is 1. For example, $3^{0} = 1$ .
</span>

Back to the Problem:
Problem 1
The x-values are in the left column. The title of the right column tells you that the function is $y = 4^{-x}$ . The x-values are:
<span>1) x = 0
</span>Plug this into $y = 4^{-x}$ to find letter a:
$y = 4^{-x}\\
y = 4^{-0}\\
y = 4^{0}\\
y = 1$
<span>
2) x = 2
</span>Plug this into $y = 4^{-x}$ to find letter b:
$y = 4^{-x}\\ 
y = 4^{-2}\\ 
y = \frac{1}{4^{2}} \\ 
y= \frac{1}{16}$
<span>
3) x = 4
</span>Plug this into $y = 4^{-x}$ to find letter c:
$y = 4^{-x}\\ 
y = 4^{-4}\\ 
y = \frac{1}{4^{4}} \\ 
y= \frac{1}{256}$
<span>

Problem 2
</span>The x-values are in the left column. The title of the right column tells you that the function is $y = (\frac{2}{3})^x$ . The x-values are:
<span>1) x = 0
</span>Plug this into $y = (\frac{2}{3})^x$ to find letter d:
$y = (\frac{2}{3})^x\\
y = (\frac{2}{3})^0\\
y = 1$
<span>
2) x = 2
</span>Plug this into $y = (\frac{2}{3})^x$ to find letter e:
$y = (\frac{2}{3})^x\\ y = (\frac{2}{3})^2\\ y = \frac{2^2}{3^2}\\
y = \frac{4}{9}$
<span>
3) x = 4
</span>Plug this into $y = (\frac{2}{3})^x$ to find letter f:
$y = (\frac{2}{3})^x\\ y = (\frac{2}{3})^4\\ y = \frac{2^4}{3^4}\\ y = \frac{16}{81}$
<span>
-------

Answers:
a = 1
b = </span> $\frac{1}{16}$ <span>
c = </span> $\frac{1}{256}$
d = 1
e = $\frac{4}{9}$
f = $\frac{16}{81}$

5 0

2 years ago

The ratio of the lengths of the sides of △ABC is 3:6:7. M, N, and K are the midpoints of the sides. Perimeter of △MNK equals 7.4

artcher [175]

Answer:

AB=2.775

BC=5.55

CA=6.475

Step-by-step explanation:

Since midpoints split their sides in half, we can see that the triangle MNK formed by the midpoints will be half the perimeter of the triangle ABC. Since P of MNK = 7.4, we know that the perimeter of ABC = 7.4 * 2, which is 14.8. Now we can split the 14.8 so that it follows the ratio.

3+6+7=16

14.8/16=0.925

AB=0.925*3=2.775

BC=0.925*6=5.55

CA=0.925*7=6.475

8 0

2 years ago

Find where the slope of the curve is defined:<br> x^2y-xy^2=4

Natali [406]

You do the implcit differentation, then solve for y' and check where this is defined.
In your case: Differentiate implicitly: 2xy + x²y' - y² - x*2yy' = 0
Solve for y': y'(x²-2xy) +2xy - y² = 0
y' = (2xy-y²) / (x²-2xy)
Check where defined: y' is not defined if the denominator becomes zero, i.e.
x² - 2xy = 0 x(x - 2y) = 0
This has formal solutions x=0 and y=x/2. Now we check whether these values are possible for the initially given definition of y:
0^2*y - 0*y^2 =? 4 0 =? 4
This is impossible, hence the function is not defined for 0, and we can disregard this.
x^2*(x/2) - x(x/2)^2 =? 4 x^3/2 - x^3/4 = 4 x^3/4 = 4 x^3=16 x^3 = 16 x = cubicroot(16)
This is a possible value for y, so we have a point where y is defined, but not y'.
The solution to all of it is hence D - { cubicroot(16) }, where D is the domain of y (which nobody has asked for in this example :-).
(Actually, the check whether 0 is in D is superfluous: If you write as solution D - { 0, cubicroot(16) }, this is also correct - only it so happens that 0 is not in D, so the set difference cannot take it out of there ...).
If someone asks for that D, you have to solve the definition for y and find that domain - I don't know of any [general] way to find the domain without solving for the explicit function).

3 0

2 years ago

The first term of a finite geometric series is 6 and the last term is 4374. The sum of all the term os 6558. find the common rat

Oliga [24]

A geometric series is written as $ar^n$ , where $a$ is the first term of the series and $r$ is the common ratio.

In other words, to compute the next term in the series you have to multiply the previous one by $r$ .

Since we know that the first time is 6 (but we don't know the common ratio), the first terms are

$6, 6r, 6r^2, 6r^3, 6r^4, 6r^5, \ldots$ .

Let's use the other information, since the last term is $4374 > 6$ , we know that $r>1$ , otherwise the terms would be bigger and bigger.

The information about the sum tells us that

$\displaystyle \sum_{i=0}^n 6r^i = 6\sum_{i=0}^n r^i = 6558$

We have a formula to compute the sum of the powers of a certain variable, namely

$\displaystyle \sum_{i=0}^n r^i = \cfrac{r^{n+1}-1}{r-1}$

So, the equation becomes

$6\cfrac{r^{n+1}-1}{r-1} = 6558$

The only integer solution to this expression is $n=6, r=3$ .

If you want to check the result, we have

$6+6*3+6*3^2+6*3^3+6*3^4+6*3^5+6*3^6 = 6558$

and the last term is

$6*3^6 = 4374$

7 0

2 years ago

Jillian’s school is selling tickets for a play. The tickets cost $10.50 for adults and $3.75 for students. The ticket sales for

Hatshy [7]

Answer:

168 adult tickets

Step-by-step explanation:

3.75(82) (82 students)

307.5

2071.5 - 307.5 ( you subtract since you only need to know the number of adults)

1764

1764/10.5 (you divide since each adult is 10.5$)

168

3 0

2 years ago

Read 2 more answers