Piecewise Notation Help

[epicjacob1123]

I have a function in piecewise notation.
f(x) = tan x for pi/4 < x < pi/2
ax + b for x less than or equal pi/4.

I have to find a and b so that the function is differentiable at x=pi/4. I understand I have to make tan (pi/4) equal to a (pi/4)+b, but I am not quite sure how to do that.

 

[Deleted member 4993]

I have a function in piecewise notation.
f(x) = tan x for pi/4 < x < pi/2
ax + b for x less than or equal pi/4.

I have to find a and b so that the function is differentiable at x=pi/4. I understand I have to make tan (pi/4) equal to a (pi/4)+b, but I am not quite sure how to do that.

tan(π/4) = ??

 

[Ishuda]

I have a function in piecewise notation.
f(x) = tan x for pi/4 < x < pi/2
ax + b for x less than or equal pi/4.

I have to find a and b so that the function is differentiable at x=pi/4. I understand I have to make tan (pi/4) equal to a (pi/4)+b, but I am not quite sure how to do that.

What is the derivative of tan(x)? Set that equal to the derivative of ax+b at x=$\displaystyle \frac{\pi}{4}$. That will give you one equation for the unknowns a and b. Set tan(x) equal to ax+b at x=$\displaystyle \frac{\pi}{4}$. That will give you another equation for the unknowns. From those two equations you should be able to find the quantities a and b.

 

[stapel]

I have a function in piecewise notation.

\(\displaystyle f(x)\, =\, \begin{cases}\tan(x)&\mbox{for }\, \frac{\pi}{4}\, <\, x\, <\, \frac{\pi}{2}\\ax\, +\, b&\mbox{for }\, x\, \leq\, \frac{\pi}{4}\end{cases}\)

f(x) = tan x for pi/4 < x < pi/2
ax + b for x less than or equal pi/4.

I have to find a and b so that the function is differentiable at x=pi/4. I understand I have to make tan (pi/4) equal to a (pi/4)+b, but I am not quite sure how to do that.

The function has to be "smooth" at the join of the two pieces. At the very least, they have to match up; they have to be equal. So do the algebra for that:

Plug in the x-value at the breakpoint (being pi/4). Evaluate each half. Set equal.

Differentiate each "half". Plug in the x-value at the breakpoints. Evaluate each half. Set equal.

This gives you a system of two equations in two variables (being "a" and "b"). Solve the system, using the methods you used back in algebra. Plug these into the original system. Graph to make sure the pieces look "smooth" at the breakpoint. (It's not a proof of your answer, but can be quite reassuring on an intuitive level.)

If you get stuck, please reply showing your work so far. Thank you!