Which of the following values are in the range of the function graph below check all that apply

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

Figure ABCD is a parallelogram. If point C lies on the line y= -1, what is the x-value or point C?

MrRa [10]

Answer:

The correct answer is option D. 4

Step-by-step explanation:

Points to remember

Opposite sides of the parallelogram are equal.

From the figure we can see that,AB and CD are opposite sides.

Therefore AB = CD

To find the length of AB

From the figure we get, AB = 4 units

Therefore CD = 4 units

To find the x coordinate of point C

y coordinate is -1

x coordinate is 4 units from point D

x coordinate of D is 0, therefore x coordinate of C = 0 + 4 = 4

The correct answer is option D. 4

4 0

2 years ago

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A coin collector is able to collect five coins each day. The collector has five coins on the first day, 10 on the second, 15 on

slavikrds [6]

5 X (number) would be it

7 0

2 years ago

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Find the complex conjugate of the given number -1+4i

adell [148]

With every imaginary term, there are 2 complex conjugates with opposing signs. So if I had the term a+bi, the complex conjugate would be a-bi. So given your example of -1+4i, the complex conjugate would just be the opposite sign, now negative, to get -1-4i.

3 0

2 years ago

Find the volume of the solid. 11 cm 11 cm 11 cm

grin007 [14]

Answer:1331cm

Step-by-step explanation:vol=lbh

11*11*11cm=1331cm

6 0

3 years ago