Problem 16 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(15-18\) , determine whether the curve has a tangent at the indicated point, If it does, give its slope, If not, explain why not. $$f(x)=\left\\{\begin{array}{ll}{-x,} & {x<0} \\ {x^{2}-x,} & {x \geq 0}\end{array}\right.\( at \)x=0$$

Step-by-Step Solution

Answer

Yes, the curve has a tangent at \(x=0\) and its slope is -1.

Step 1: Evaluate the Function Value at \(x=0\)

The function \(f(x)\) is defined as \(-x\) for \(x < 0\) and as \(x^2 - x\) for \(x \geq 0\). Since \(x = 0\) falls under the second category, it's necessary to evaluate the value of \(x^2 - x\) at \(x = 0\) which gives us 0.

Step 2: Determine the Derivative of \(f(x)\) on both Intervals

To check whether \(f(x)\) has a tangent at \(x=0\), we must also check the behavior of the function on either side of this point by examining the derivatives for \(x < 0\) and \(x \geq 0\). The derivative of \(-x\) is \(-1\) for \(x < 0\), and the derivative of \(x^2 - x\) is \(2x - 1\) for \(x \geq 0\).

Step 3: Evaluate the Derivative Value at \(x=0\)

With the derivative functions, we evaluate their values at \(x = 0\). For \(x < 0\), the derivative is -1; for \(x \geq 0\), the derivative is -1. The derivative of our function does exist on both sides and is equal.

Step 4: Conclude whether the function has a Tangent at \(x=0\)

As the function's value and the derivative at \(x=0\) are equal from both sides, it means that the function is differentiable at this point and hence the curve has a tangent at \(x=0\). The slope of the tangent is equal to the derivative at this point, which is -1.

Key Concepts

Differentiability

Differentiability is a key concept when we want to determine if a curve has a tangent at a specific point. In simple terms, a function is differentiable at a point if it has a well-defined derivative at that point. This means that not only does the function have a derivative, but that derivative is consistent as you approach the point from either side. Differentiability implies continuity, however, the opposite is not always true.

For our exercise, we are interested in determining differentiability at the point where \(x = 0\). By evaluating both sides of the function (for \(x < 0\) and \(x \geq 0\)), we examine if the function behaves smoothly around that point. Since the derivatives on either side match, we conclude that the function is indeed differentiable at \(x = 0\), allowing us to define a tangent line with a specific slope there.

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the domain. Think of it as a set of rules where different conditions apply to different values of \(x\).

In our exercise, the function is defined differently for \(x < 0\) and \(x \geq 0\):

• When \(x < 0\), the function is \(f(x) = -x\).
• When \(x \geq 0\), the function changes to \(f(x) = x^2 - x\).

These definitions make it important to evaluate the function and its derivatives separately within these regions. For a piecewise function to have a tangent at a specific point, the function must be continuous and share the same derivative on both sides of the point. This requirement is crucial for understanding piecewise functions in calculus.

Derivative Evaluation

Derivative evaluation is an essential method for understanding the rate at which functions change. To find if a function has a tangent line, we calculate its derivative. The derivative tells us the slope of the tangent at any given point.

For the given exercise, derivative evaluation involves looking at each piece of the piecewise function:

• The derivative of \(-x\) for \(x < 0\) is \(-1\).
• The derivative of \(x^2 - x\) for \(x \geq 0\) is \(2x - 1\).

At the point \(x = 0\), we evaluate \(2x - 1\) which gives us \(-1\). The fact that both parts of the function yield the same derivative value ensures that the tangent's slope is consistent from both sides at \(x = 0\). Therefore, the function is differentiable here, confirming the existence of a tangent with a slope of \(-1\).