# Find k for Continuous & Differentiable Function

• golb0016
In summary, to find k for the given function to be both continuous and differentiable, you need to check that the function is continuous at x=1 and that the slope at x=1 matches up for both parts of the piecewise function. By computing the limit and checking the slopes, it is determined that k=2 will make the function continuous and differentiable at x=1.
golb0016

## Homework Statement

find k for the function so it is continuous and differentiable.
x^2-1 x<=1
k(x-1) x>1

## The Attempt at a Solution

k(x-1)=0 for x=1
k(0)=0
k = 0/0?

How do I know if the function is differentiable?

Work on making it differentiable at x = 1 first - if you can, then you know it will be continuous there. (what are the left / right-hand derivatives)

Actually, that function is continuous for any value of k. The only point you have to worry about, both for continuity and differentiability, is the point x=1. You can explicitly show that the function is continuous at 1 by computing the limit of f as x approaches 1 and show that it equals f(1). You'll see it doesn't matter what k is for continuity.

To worry about differentiability, all you need to check is that the slope of the function k(x-1) at x=1 matches up with the slope of x2-1 at x=1. Remember that for a function to be differentiable at a point, the limit of the difference quotient at that point must exist. By checking that the slopes match up, you are checking that the left and right handed limits of the difference quotient equal each other (i.e., checking that the limit exists). The function is differentiable everywhere else since it is defined there by polynomials.

f(x) = x^2-1
f'(1) = 2(1) = 2

f(x) = k(x-1)
f'(1) = k = 2

Therefore k=2 and is differentiable at this point now, correct?

## 1. What is the purpose of finding k for a continuous and differentiable function?

The value of k in a continuous and differentiable function is important because it represents the slope of the tangent line at any point on the function's graph. This slope can give us information about the rate of change of the function and can help us solve optimization problems.

## 2. How do you find the value of k for a continuous and differentiable function?

To find the value of k, we can use the derivative of the function. The derivative, denoted by f'(x), represents the slope of the tangent line at any point on the function's graph. By setting f'(x) equal to k, we can solve for the value of x. This value of x represents the point on the graph where the slope is equal to k.

## 3. Can a continuous and differentiable function have more than one value of k?

Yes, it is possible for a continuous and differentiable function to have multiple values of k. This can happen when the function has a point of inflection, where the concavity changes from positive to negative or vice versa. In this case, there will be two values of k, one for each portion of the graph where the concavity changes.

## 4. How does changing the value of k affect the graph of a continuous and differentiable function?

Changing the value of k will affect the slope of the function at different points on the graph. If k is positive, the function will have a positive slope, and if k is negative, the function will have a negative slope. This will result in the graph of the function being steeper or shallower depending on the value of k.

## 5. What other applications does finding k in a continuous and differentiable function have?

Finding k in a continuous and differentiable function has many applications in real life. It can be used to model and analyze various phenomena such as population growth, financial investments, and chemical reactions. It is also used in physics to calculate the velocity and acceleration of moving objects.

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