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3 years ago

Use the distributive property to write an expression that is equivalent to 8(7+ X).

Mathematics

1 answer:

motikmotik3 years ago

5 0

Answer:

56 + 8x

Step-by-step explanation:

8(7) + 8(x)

= 56 + 8x

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A curve has equation y=x√x.find the equation of the tangent to the curve at the point(1, 5)

rewona [7]

Answer:

y - 5 = $\frac{3}{2}$ (x - 1)

Step-by-step explanation:

Note that $\frac{dy}{dx}$ = $m_{tangent}$

Differentiate using the power rule

$\frac{d}{dx}$ (a $x^{n}$ ) = na $x^{n-1}$

Given

y = x $\sqrt{x}$ = x. $x^{\frac{1}{2} }$ = $x^{\frac{3}{2} }$ , then

$\frac{dy}{dx}$ = $\frac{3}{2}$ $x^{\frac{1}{2} }$

When x = 1

$\frac{dy}{dx}$ = $\frac{3}{2}$ . 1 = $\frac{3}{2}$

The equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

Here m = $\frac{3}{2}$ and (a, b) = (1, 5), thus

y - 5 = $\frac{3}{2}$ (x - 1) ← equation of tangent

3 0

3 years ago

Find a solution to the initial value problem, y′′+18x=0,y(0)=5,y′(0)=1.

Serga [27]

We want to find a solution to the initial value problem:

$y'' + 18x = 0 \qquad,\qquad y(0) = 5 \qquad,\qquad y'(0)=1.$

We can start by integrating the equation once:

$\dfrac{\textrm{d}^2 y}{\textrm{d}x^2} + 18 x = 0 \iff \dfrac{\textrm{d}^2 y}{\textrm{d}x^2} = -18 x \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -18\displaystyle\int x\textrm{ d}x \iff \dfrac{\textrm{d}y}{\textrm{d}x}=-18\dfrac{x^2}{2} + C \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + C.$

Using the initial condition $y'(0) = 1$ , we can determine the integration constant $C$ :

$\dfrac{\textrm{d}y}{\textrm{d}x}\Big\vert_{x= 0} = 1 \iff -9 \times 0^2 + C = 1 \iff C = 1.$

Therefore, we have:

$\dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + 1$

We can now integrate again:

$y(x) = \displaystyle\int\dfrac{\textrm{d}y}{\textrm{d}x}\textrm{ d}x = \int\left(-9x^2+1\right)\textrm{d}x = -9\int x^2\textrm{ d}x + \int\textrm{d}x =\\\\= -9\dfrac{x^3}{3} + x + K = -3x^3 + x + K.$