I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT

3. The curve C with equation y=f(x) is such that, dy/dx = 3x^2 + 4x +k

Andreas93 [3]

a. Given that y = f(x) and f(0) = -2, by the fundamental theorem of calculus we have

$\displaystyle \frac{dy}{dx} = 3x^2 + 4x + k \implies y = f(0) + \int_0^x (3t^2+4t+k) \, dt$

Evaluate the integral to solve for y :

$\displaystyle y = -2 + \int_0^x (3t^2+4t+k) \, dt$

$\displaystyle y = -2 + (t^3+2t^2+kt)\bigg|_0^x$

$\displaystyle y = x^3+2x^2+kx - 2$

Use the other known value, f(2) = 18, to solve for k :

$18 = 2^3 + 2\times2^2+2k - 2 \implies \boxed{k = 2}$

Then the curve C has equation

$\boxed{y = x^3 + 2x^2 + 2x - 2}$

b. Any tangent to the curve C at a point (a, f(a)) has slope equal to the derivative of y at that point:

$\dfrac{dy}{dx}\bigg|_{x=a} = 3a^2 + 4a + 2$

The slope of the given tangent line $y=x-2$ is 1. Solve for a :

$3a^2 + 4a + 2 = 1 \implies 3a^2 + 4a + 1 = (3a+1)(a+1)=0 \implies a = -\dfrac13 \text{ or }a = -1$

so we know there exists a tangent to C with slope 1. When x = -1/3, we have y = f(-1/3) = -67/27; when x = -1, we have y = f(-1) = -3. This means the tangent line must meet C at either (-1/3, -67/27) or (-1, -3).

Decide which of these points is correct:

$x - 2 = x^3 + 2x^2 + 2x - 2 \implies x^3 + 2x^2 + x = x(x+1)^2=0 \implies x=0 \text{ or } x = -1$

So, the point of contact between the tangent line and C is (-1, -3).

7 0

2 years ago

Chris made his aunt a garden. The width is 7 2/5 ft and the length is 5 1/2 ft. What is the area of the garden?

pishuonlain [190]

Answer:

Hello! if Chris garden is a square shape then the answer is 40 7/10 in fraction form or 40.7 in decimal form

6 0

2 years ago

Y = 4(x–3) + 7<br> What is the Horizontal<br> Asymptote?

Tema [17]

Answer:

y = 4(x-3) + 7

4x-3 + 7

+3. +3

_____________

4x. 10

4. 4

_______________

x = 2.5

Horizontal Asymptote are horizontal lines the graph approaches

3 0

3 years ago

I need the steps to this, please. It says solve for x. Each figure is a parallelogram. Show your work.

Trava [24]

With any parallelogram, the diagonals bisect each other. This is another way of saying that they cut each other in half.

FH is one diagonal that is split into two equal pieces by the other diagonal EG.

The two parts of FH (KH and KF) are congruent to each other, so KH = KF. They combine back to FH

By the segment addition postulate
KH + KF = FH
KH + KH = FH .... KF has been replaced with KH (works because KF = KH)
2*KH = FH

Now use substitution
2*KH = FH
2*15 = FH .... replace KH with 15 (since KH = 15)
2*15 = 4x-2 ... replace FH with 4x-2 (since FH = 4x-2)

and solve for x
2*15 = 4x-2
30 = 4x-2
30+2 = 4x-2+2 ... add 2 to both sides
32 = 4x
4x = 32
4x/4 = 32/4 ... divide both sides by 4
x = 8

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Answer: x = 8

5 0

3 years ago

Marking brainliest

sertanlavr [38]

Answer:

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5 0

3 years ago