Question

Help with a differentiable function problem? Will medal!

Answer 1

\[f(x)=asin(x)+b\] where x<=2pi
\[f(x)=x²-pix+2\] where x>2pi
What are values a and b that f(x) is differentiable?

Answer 2

That is supposed to be x^2 - pi(x) +2 in the second equation

Answer 3

@jim_thompson5910 any chance you could help?

Answer 4

what do you get when you differentiate each function?

Answer 5

acosx and 2x-pi

Answer 6

plug x = 2pi into each derivative function and then set the two equal to one another, and solve for a

Answer 7

I'm going to do a bit of notation change

Answer 8

Let

g(x) = a*sin(x) + b
h(x) = x^2 - pi*x + 2

Answer 9

That makes so much sense! And that isn't a problem that only one really touches 2pi?

Answer 10

Now let f(x) be the piecewise function composed of g(x) and h(x) (based on the restrictions above)

Answer 11

if f(x) is differentiable, then it has to be continuous

so that means g(2pi) = h(2pi) has to be true since this is the only possible discontinuity (the junction where they join)

Answer 12

also if f(x) is differentiable, then g ' (2pi) = h ' (2pi) also has to be true

Answer 13

Oh, of course! so because f(x) is differentiable, the derivatives will have the same slope at the point which allows me just to substitute and solve?

Answer 14

Would I then get b based off of the y-value of h(x) at x=2pi?

Answer 15

what do you get for the value of 'a'?

Answer 16

about 9.4?

Answer 17

you should have a = 3pi

Answer 18

so you have it roughly

Answer 19

g(x) = a*sin(x) + b turns into g(x) = 3pi*sin(x) + b

Answer 20

we have this now

g(x) = 3pi*sin(x) + b
h(x) = x^2 - pi*x + 2

Answer 21

that makes sense

Answer 22

to find b, you need to plug in x = 2pi into the original g and h, set the two equal to one another, and solve for b

Answer 23

g(x) = h(x)

g(2pi) = h(2pi)

3pi*sin(2pi) + b = (2pi)^2 - pi*2pi + 2

solve for b

Answer 24

And they will equal each other because they will be the same y value for x=2pi?

Answer 25

yeah we want to force the two functions to line up and not have a jump discontinuity there

Answer 26

Wow jim, good understanding of functions, differentiation and trig obvious!

Answer 27

thanks

Answer 28

So b= approx. 21.74?

Answer 29

that looks about right, I'm getting (2pi)^2 - pi*2pi + 2 = 21.7392 which rounds to 21.74

Answer 30

if you need the exact value of b, then

(2pi)^2 - pi*2pi + 2 = 4pi^2 - 2pi^2 + 2 = 2pi^2 + 2

Answer 31

just curious: is a.sinx = a. dy(sinx)/dx?

Answer 32

@jim_thompson5910

Answer 33

what do you mean? you mean differentiating a*sin(x) ?

Answer 34

yes, just multiplying by constant a jim?

Answer 35

oh, yeah 'a' is a constant

Answer 36

so

\[\Large \frac{d}{dx}[a*\sin(x)] = a*\frac{d}{dx}[\sin(x)]\]

Answer 37

and there you go @TSV , that is the rule

Answer 38

Thank you so much! This all made a lot more sense after walking through some steps. I really appreciate it!

Answer 39

oh well, @jim_thompson5910 is really good!

Answer 40

you're welcome