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2 years ago

Given the relation y = –4x2 – 8, if the input is 2, what is the output? 8 –8 –16 –24

Mathematics

1 answer:

Pani-rosa [81]2 years ago

3 0

Y = -4x² - 8
y = -4(2)² - 8
y = -4(4) - 8
y = -16 - 8
y = -24

The output is -24.

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Find the integral using substitution or a formula.

Nadusha1986 [10]

$\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx$

Derivative of the denominator:
$\rm (x^2+2x+5)'=2x+2$

Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

$\rm \int \dfrac{x^2+2+5}{x^2+2x+5}~dx$

and then split the fraction,

$\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx$

Based on our previous test, we know that a simple substitution will work for the first integral: $\rm \quad u=x^2+2x+5\qquad\to\qquad du=2x+2~dx$

So the first integral changes,

$\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx$

integrating to a log,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx$

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

$\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2$

So we have this going on,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx$

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx$

Bringing all that stuff together as a single square,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx$

Making the substitution: $\rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx$

giving us,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du$

simplying a lil bit,

$\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du$

and hopefully from this point you recognize your arctangent integral,

$\rm ln|x^2+2x+5|+\frac52arctan(u)$

undo your substitution as a final step,
and include a constant of integration,

$\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c$

Hope that helps!
Lemme know if any steps were too confusing.

8 0

3 years ago

Determine the formula for the nth term of the sequence:<br>-2,1,7,25,79,...

rodikova [14]

A plausible guess might be that the sequence is formed by a degree-4* polynomial,

$x_n = a n^4 + b n^3 + c n^2 + d n + e$

From the given known values of the sequence, we have

$\begin{cases}a+b+c+d+e = -2 \\ 16 a + 8 b + 4 c + 2 d + e = 1 \\ 81 a + 27 b + 9 c + 3 d + e = 7 \\ 256 a + 64 b + 16 c + 4 d + e = 25 \\ 625 a + 125 b + 25 c + 5 d + e = 79\end{cases}$

Solving the system yields coefficients

$a=\dfrac58, b=-\dfrac{19}4, c=\dfrac{115}8, d = -\dfrac{65}4, e=4$

so that the n-th term in the sequence might be

$\displaystyle x_n = \boxed{\frac{5 n^4}{8}-\frac{19 n^3}{4}+\frac{115 n^2}{8}-\frac{65 n}{4}+4}$

Then the next few terms in the sequence could very well be

$\{-2, 1, 7, 25, 79, 208, 466, 922, 1660, 2779, \ldots\}$

It would be much easier to confirm this had the given sequence provided just one more term...

* Why degree-4? This rests on the assumption that the higher-order forward differences of $\{x_n\}$ eventually form a constant sequence. But we only have enough information to find one term in the sequence of 4th-order differences. Denote the k-th-order forward differences of $\{x_n\}$ by $\Delta^{k}\{x_n\}$ . Then

• 1st-order differences:

$\Delta\{x_n\} = \{1-(-2), 7-1, 25-7, 79-25,\ldots\} = \{3,6,18,54,\ldots\}$

• 2nd-order differences:

$\Delta^2\{x_n\} = \{6-3,18-6,54-18,\ldots\} = \{3,12,36,\ldots\}$

• 3rd-order differences:

$\Delta^3\{x_n\} = \{12-3, 36-12,\ldots\} = \{9,24,\ldots\}$

• 4th-order differences:

$\Delta^4\{x_n\} = \{24-9,\ldots\} = \{15,\ldots\}$

From here I made the assumption that $\Delta^4\{x_n\}$ is the constant sequence {15, 15, 15, …}. This implies $\Delta^3\{x_n\}$ forms an arithmetic/linear sequence, which implies $\Delta^2\{x_n\}$ forms a quadratic sequence, and so on up $\{x_n\}$ forming a quartic sequence. Then we can use the method of undetermined coefficients to find it.

5 0

2 years ago

Which expressions are equivalent to -2.5(1-2n) - 1.5n

Keith_Richards [23]

Answer:

-5n - 4

Step-by-step explanation:

The goal is to simplify this expression.

1. Distribute

-2.5 * 1 = -2.5

-2.5 * 2n = -5n

Now, we have

-2.5 - 5n - 1.5

2. Combine like terms

-2.5 + (-1.5) = -4 (+ * - = - The - cancels out the +)

3. Put it together

-5n - 4

I hope I helped!

Let me know if you need anything else!

~ Zoe

4 0

2 years ago

Allen is buying a video that originally cost $49.He has a 10%-off coupon and will have to pay a 6% sales which expression will c

Gekata [30.6K]

Answer:

$46.746

Step-by-step explanation:

Allen is buying a video that originally cost = $ 49

Discount coupon = 10%

Sales tax= 6%

Hence, the price of video after discount coupon and sales tax= $ 49 × $\frac{100-10}{100}$ × $\frac{100+6}{100}$

Hence, the price of video after discount coupon and sales tax= $ 49 × $\frac{90}{100}$ × $\frac{106}{100}$

Price of video after discount coupon and sales tax= $ 49 × 0.9× 1.06

price of video after discount coupon and sales tax= $46.746

Hence, the correct answer is $46.746

5 0

3 years ago

sarah meeham blends coffee for tasti delight. she needs to prepare 160 pounds of blended coffee beans selling for $4.97 per poun

Viefleur [7K]

Given:
h = high quality bean ; costs 6 per pound
c = cheaper bean ; costs 3.25 per pound

160 pounds of blended coffee beans at 4.97 per pound

h + c = 160

6h + 3.25c = 160(4.97)
6h + 3.25c = 795.20

h = 160 - c

6(160-c) + 3.25c = 795.20
960 - 6c + 3.25c = 795.20
- 2.75c = 795.20 - 960
- 2.75c = -164.80
c = - 164.80 / -2.75
c = 59.92 or 60 pounds

h = 160 - c
h = 160 - 60
h = 100 pounds

Sarah must blend 100 pounds of high quality bean and 60 pounds of cheaper bean.

3 0

3 years ago