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3 years ago

14

What is the range of the function f(x) = 3x2 + 6x – 8?

1 answer:

Sav [38]3 years ago

5 0

Find minimum and max y value because that is what range is
we see it opens up becasue 3 is positive
x value of vertex is -b/2a when you have f(x)=ax^2+bx+c

f(x)=3x^2+6x-8
vertex is -6/(2*3)=-6/6=-1
find y value by subsituiong it for x
f(-1)=3(-1)^2+6(-1)-8
f(-1)=3(1)-6-8
f(-1)=3-14
f(-1)=-11

range is from -11 to infinity

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Consider the initial value problem y′+5y=⎧⎩⎨⎪⎪0110 if 0≤t<3 if 3≤t<5 if 5≤t<[infinity],y(0)=4. y′+5y={0 if 0≤t<311 i

rosijanka [135]

It looks like the ODE is

$y'+5y=\begin{cases}0&\text{for }0\le t$

with the initial condition of $y(0)=4$ .

Rewrite the right side in terms of the unit step function,

$u(t-c)=\begin{cases}1&\text{for }t\ge c\\0&\text{for }t$

In this case, we have

$\begin{cases}0&\text{for }0\le t$

The Laplace transform of the step function is easy to compute:

$\displaystyle\int_0^\infty u(t-c)e^{-st}\,\mathrm dt=\int_c^\infty e^{-st}\,\mathrm dt=\frac{e^{-cs}}s$

So, taking the Laplace transform of both sides of the ODE, we get

$sY(s)-y(0)+5Y(s)=\dfrac{e^{-3s}-e^{-5s}}s$

Solve for $Y(s)$ :

$(s+5)Y(s)-4=\dfrac{e^{-3s}-e^{-5s}}s\implies Y(s)=\dfrac{e^{-3s}-e^{-5s}}{s(s+5)}+\dfrac4{s+5}$

We can split the first term into partial fractions:

$\dfrac1{s(s+5)}=\dfrac as+\dfrac b{s+5}\implies1=a(s+5)+bs$

If $s=0$ , then $1=5a\implies a=\frac15$ .

If $s=-5$ , then $1=-5b\implies b=-\frac15$ .

$\implies Y(s)=\dfrac{e^{-3s}-e^{-5s}}5\left(\frac1s-\frac1{s+5}\right)+\dfrac4{s+5}$

$\implies Y(s)=\dfrac15\left(\dfrac{e^{-3s}}s-\dfrac{e^{-3s}}{s+5}-\dfrac{e^{-5s}}s+\dfrac{e^{-5s}}{s+5}\right)+\dfrac4{s+5}$

Take the inverse transform of both sides, recalling that

$Y(s)=e^{-cs}F(s)\implies y(t)=u(t-c)f(t-c)$

where $F(s)$ is the Laplace transform of the function $f(t)$ . We have

$F(s)=\dfrac1s\implies f(t)=1$

$F(s)=\dfrac1{s+5}\implies f(t)=e^{-5t}$

We then end up with

$y(t)=\dfrac{u(t-3)(1-e^{-5t})-u(t-5)(1-e^{-5t})}5+5e^{-5t}$

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4 years ago

What is the y-coordinate of the point that divides the

Nina [5.8K]

Answer:

5

Step-by-step explanation:

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3 years ago

Please help 20 points geometry triangle questions

slava [35]

Answer:

1. yes, SAS

2. yes, ASA

3. yes, SSA

4. yes, SAS

5. no

6. yes, HL and SSA

7. yes, HL and ASA

8. yes, ASA

9. no

10. yes, SSS

11. yes, ASA

12. yes, HL and SSA

13. yes, SAS

14. no

15. no

16. no

17. yes, ASA

18. yes, ASA

19. yes, ASA

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3 years ago