Problem 15 - 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}{2-2 x-x^{2},} & {x<0} \\ {2 x+2,} & {x \geq 0}\end{array}\right.\( at \)x=0$$

Step-by-Step Solution

Answer

No, the function \(f(x)\) does not have a tangent at \(x=0\), because the slopes of the tangent lines to the two pieces of the function do not match at this point.

Step 1: Identify the domains and the corresponding functions

First, notice that the function \(f(x)\) is defined by two equations. The first equation, \(2-2x-x^{2}\), applies when \(x\) is less than zero. The second equation, \(2x+2\), applies when \(x\) is greater than or equal to zero.

Step 2: Calculate the derivative for each piece of the function

To find the slope of the tangent line to the curve at any point, we will need to find the derivative of each piece of the function. For the first one, \(-2-2x\), and for the second one, \(2\). These are the slopes of the tangent lines to the curves defined by each piece of the function.

Step 3: Compare the two derivatives at x=0

To determine if the function has a tangent line at \(x=0\), we need to see if the two piecewise functions have the same derivative at this point. Substituting \(x=0\) into the first derivative gives us \(-2\), and the second derivative is just \(2\). Since the two derivatives do not match, we conclude that the function does not have a tangent at \(x=0\).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different formulas based on the value of the independent variable. In calculus, understanding piecewise functions is crucial as they can model complex, real-world situations that have different rules for different scenarios.

For example, a company might charge different rates depending on the amount of goods produced or the distance shipped. In our exercise, the function in question, represented as a piecewise function, changes its formula at the point where the value of the independent variable, x, is zero. One formula is used for x<0 and another for x≥0. This abrupt change can potentially affect the continuity and differentiability of the function, which are properties that are essential for calculating the tangent line at a particular point.

Derivative Calculation

In calculus, the derivative represents the rate at which a function's output changes with respect to changes in its input. Thus, it's a measure of how a function reacts to small changes in its variables. In practical terms, it can represent things such as speed or rate of change of cost.

To calculate the derivative of piecewise functions, you need to derive each piece individually. You must consider the domain of each piece when taking the derivative. The key fact is that each piece's derivative provides the slope of the tangent line to that part of the function's curve. In the exercise, for each domain, we calculated two different derivatives, which indicate two different slopes, responding to how the function behaves on either side of x=0.

Slope of a Tangent

When talking about the slope of a tangent line, you're essentially seeking the instantaneous rate of change of the function at a certain point. In simpler terms, it is like measuring the steepness of the hill at precisely the point where you stand.

To find the slope of a tangent line to a curve at a specific point, you calculate the derivative of the function at that point. The derivative, when evaluated at a particular x value, tells you the slope of the tangent line to the curve at that point. However, if you're dealing with piecewise functions, you must ensure the slopes from different pieces agree at the junction point, like at x=0 in our exercise, to have a single, well-defined tangent line.

Continuity and Differentiability

Two fundamental properties in calculus are continuity and differentiability. A function is continuous at a point if there is no interruption in the curve at that point—the function's value approaches and hits a particular y-value as x approaches the point from either side.

For differentiability, a function is considered differentiable at a point if it has a derivative there. If a curve has a sharp corner or cusp at that point, like it might in piecewise functions, then the function is not differentiable there because the slope cannot be defined. Our exercise showcases this as the function has two different slopes at the join point of the pieces, indicating a lack of differentiability at x=0 and consequently, a lack of a well-defined tangent.