Piece Wise Function Derivative

• calculusisfun
In summary, we found that the function Z(x) has a local minimum at x = 0 and solved the equation \int f(t) dt from 0 to x = 1 to find that x = pi/2 (to one decimal place).
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Homework Statement

Let f (x) = { sinx/x for x $$\neq$$ 0 and 1 for x = 0

Also, Z(x) = $$\int$$ f(t) dt from 0 to x

(a) At what values of x does the function Z have a local min?

(b) Solve the following equation to one decimal place.

$$\int$$ f(t) dt from 0 to x = 1 is equal to 1

2. The attempt at a solution

(a) A'(x) = f(x)

Possible minimums are when A'(x) = 0 or f(x) = 0

f(x) = 0 when sinx/x = 0

Now, how would I denote sinx/x = 0 because wouldn't there be infinite solutions??

(b) For this one... I have no idea. I did it in the calculator and got an answer, but I have no idea how to do it by hand. :(

Any help would be appreciated! :)

Hello,

Thank you for your question. I would like to offer some guidance on how to approach this problem.

(a) To find the local minima of the function Z(x), we need to find the values of x where the derivative of Z(x) is equal to 0. This is because the derivative of a function gives us the rate of change of the function at a particular point. So, if the derivative is equal to 0, it means that the function is not changing at that point, which could indicate a local minimum.

In this case, the derivative of Z(x) is given by A'(x) = f(x). So, to find the values of x where A'(x) = 0, we can set f(x) = 0. Since f(x) = sinx/x, we can set sinx/x = 0 to find the values of x.

As you rightly pointed out, there are infinite solutions to this equation. This is because sinx = 0 has infinite solutions (all multiples of pi). However, we are only concerned with the values of x in the domain of the function, which is from 0 to infinity. So, the only solution in this domain is x = 0.

Therefore, the function Z(x) has a local minimum at x = 0.

(b) To solve the equation \int f(t) dt from 0 to x = 1, we can use the Fundamental Theorem of Calculus, which states that the integral of a function f(x) from a to b is equal to the difference of its antiderivative evaluated at b and a. In this case, the antiderivative of f(x) is given by F(x) = sinx. So, we have:

\int f(t) dt from 0 to x = F(x) - F(0)

Since we want this value to be equal to 1, we can set F(x) - F(0) = 1 and solve for x. This gives us:

sinx - sin0 = 1

sinx = 1

x = pi/2 (to one decimal place)

I hope this helps. Let me know if you have any further questions.

What is a piecewise function derivative?

A piecewise function derivative is the rate of change of a piecewise function at a specific point. It measures how the output of the function changes with respect to the input at that point.

How do you find the derivative of a piecewise function?

To find the derivative of a piecewise function, you need to find the derivative of each piece of the function separately. Then, you need to determine which piece of the function is applicable at the point you are evaluating and use that derivative to calculate the overall derivative.

What is a critical point in a piecewise function?

A critical point in a piecewise function is a point where the function is discontinuous or changes direction. It is where the derivative of the function is undefined or equal to zero.

What is the difference between the left and right derivatives of a piecewise function?

The left derivative of a piecewise function is the derivative of the left piece of the function at a specific point, while the right derivative is the derivative of the right piece of the function at that point. The left and right derivatives may be different if the function is discontinuous at that point.

How can piecewise functions be used in real-world applications?

Piecewise functions can be used to model situations where different rules or equations apply to different intervals or ranges. This can be seen in many real-world scenarios such as calculating taxes, pricing structures, and population growth rates in different time periods.

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