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lesantik [10]
2 years ago
13

Find the values of m and b that make the following function differentiable.

Mathematics
1 answer:
dimaraw [331]2 years ago
4 0
Now, bearing in mind that, for a graph, to be "differentiable", means a "smooth transition", not an abrupt edge or a "cusp", the first subfunction of the piece-wise is just a quadratic, the second subfunction is a linear, well, so the linear needs to hit the quadratic "smoothly", not make an abrupt edge, for that, the linear must continue the quadratic from 2 onwards, since 2 is the range edge.

now, what is the piece-wise when x = 2? well, f(2) = x² or (2)² or 4

that means, the linear needs to hit it at that point, to make it so, let's make it the slope of the linear, the same as the quadratic's.

what's the quadratic's slope? well, simple enough

\bf \left. \cfrac{dy}{dx}=2x \right|_{x=2}\implies 4\impliedby \textit{this means }
\begin{array}{llll}
mx+b\\
\uparrow \\
4
\end{array}

so we have y = 4x + b

now, recall, for the quadratic, when x =2, y = 4, so for the linear, when x = 2, y = 4 as well, this means

\bf 4 = 4(2) + b\implies 4=8+b\implies 4-8=b\implies \boxed{-4=b}
\\\\\\
mx+b\implies 4x-4\\\\
-------------------------------\\\\
f(x)=
\begin{cases}
x^2&x\le 2\\
4x-4&x\ \textgreater \ 2
\end{cases}

check the picture below.

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What is the quotient of 4.38 and 0.5
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2 years ago
Can anyone help me with both problems please?
Eva8 [605]

Answer:

1.y=2x+7

2.y=1/2x+2

Step-by-step explanation:

Graph 1-

The slope-intercept form is y=mx+b

b= y-intercept which in graph 1 is 7

so y=mx+7

in order to find m you use the y2-y1/x2-x1 equation.

This means find two coordinates on the graph such as (0,7) and (-1,5).

Now find y2:5

y1:7

x2:-1

x1:0

y2-y1= 5-7=-2

x2-x1= -1

That gives you -2/-1 which you then simplify to = 2

So the graph would be y=2x+7

Graph 2:

You will do the same steps: Find where the line passes through y on a graph, which will be y=2.

b=2

y=mx+2

then find two coordinates, (0,2)(2,3)

y2=3

y1=2

x2=2

x1=0

y2-y1=3-2=1

x2-x1=2-0=2

The slope of the graph which is m= 1/2

y=1/2x+2

8 0
2 years ago
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