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Tasya [4]
4 years ago
13

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

Mathematics
1 answer:
user100 [1]4 years ago
7 0
Let g(x) = x^2 and h(x) = mx+b
The piecewise function f(x) is basically a combination of g(x) and h(x) depending on what x is. 

If x is equal to 3 or smaller, then f(x) = g(x) = x^2
If x is larger than 3, then f(x) = h(x) = mx+b

Plug in x = 3 into g(x)
g(x) = x^2
g(3) = 3^2
g(3) = 9
And do the same for h(x)
h(x) = mx+b
h(3) = m*3+b
h(3) = 3m+b

In order for f(x) to be differentiable at x = 3, the two functions g(x) and h(x) must meet up at this x value. The function f(x) must be continuous at x = 3. In other words, g(x) = h(x) must be true when x = 3

So we can equate the two function outputs
g(3) = h(3)
9 = 3m+b
Solve for b to get
b = 9-3m

---------------------------------------------------------------
Now differentiate each function g(x) and h(x) with respect to x. Plug in x = 3 after you differentiated
g(x) = x^2
g ' (x) = 2x
g ' (3) = 2*3
g ' (3) = 6
h(x) = mx+b
h ' (x) = m
h ' (3) = m

If f(x) is differentiable at x = 3, then f ' (x) must be continuous at x = 3

This means,
g ' (x) = h ' (x)
g ' (3) = h ' (3)
6 = m
m = 6

Now use this value of m to find b
b = 9 - 3m
b = 9 - 3*6
b = 9 - 18
b = -9
---------------------------------------------------------------

In summary, we found that
m = 6
b = -9
which are the values needed to make f(x) differentiable
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Sergio [31]

Answer:

The explicit function is:

S_{n} =Pz^{n-1} +C(\frac{z^{n-1}-1}{z-1} )

where

z=(1+\frac{0.02}{12})

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S_{12} = $1211.06

Step-by-step explanation:

Expanding a few steps of the compound interest:

S_{1}=100\\S_{2}=S_{1}(1+\frac{0.02}{12})+100\\S_{3}=S_{2}(1+\frac{0.02}{12})+100\\...\\S_{n}=S_{n-1}(1+\frac{0.02}{12})+100

We can write:

(1+\frac{0.02}{12})=z

100=C for deposits

Then, expanding the previous equations would yield:

S_{1}=C\\S_{2}=Cz + C\\S_{3}=Cz^{2}+Cz + C\\S_{n}=Cz^{n-1} +...+Cz+C

This is a geometric series form, which can be simplified to:

S_{n}=C(\frac{z^{n}-1}{z-1} )

plugin in the values for the 12 month gives:

S_{12}=100(\frac{(1+\frac{0.02}{12})^{12} -1}{\frac{0.02}{12}})=1211.0613

6 0
3 years ago
As an estimation we are told 5 miles is 8 km. <br> Convert 70 miles to km. <br> km
Lena [83]

Answer:  112 km

=====================================================

Work Shown:

(5 miles)/(8 km) = (70 miles)/(x km)

5/8 = 70/x

5*x = 8*70 ... cross multiply

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x = 560/5 .... divide both sides by 5

x = 112

According to the given estimation, 70 miles converts to 112 km.

The more accurate conversion is approximately 70 miles = 112.65408 km, so this estimation isn't too far off.

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4 years ago
If Nathan begins his journey after 9 a.m. he is allowed a discount
Anuta_ua [19.1K]

Answer:

a) The discounted prince is 70% of the full price.

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Step-by-step explanation:

a) If the full price is 100%, then you subtract 30% which is the discount and you get 70 %.

b) 22.75 is 70% of 32.5

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There are 19 men and 31 women on an airplane. What is the ratio of the number of male passengers to the total number of passenge
castortr0y [4]

Answer:

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Step-by-step explanation:

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3 0
2 years ago
Read 2 more answers
2x - y = 3 show the work and a graph
Galina-37 [17]
First of all, if we want to graph this we will need to make it into slope intercept form

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add y to both sides
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we now know that the y-intercept is at (0, -3)
using that we plot the first point
The slope is 2 
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using that we make a line

(I'm sure you can graph it ;D)

Hope this helps :)

7 0
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