Question

In Exercises 44 and \(45,\) let $$f(x)=\left\\{\begin{array}{ll}{2 x+1,} & {x \leq 0} \\ {x^{2}+1,} & {x > 0}\end{array}\right.$$ Multiple Choice Which of the following is equal to the lefthand derivative of \(f\) at \(x=0 ? \quad\) \(\begin{array}{llll}{\text { (A) } 2 x} & {\text { (B) } 2} & {\text { (C) } 0} & {\text { (D) }-\infty}\end{array}\) \((\mathbf{E}) \infty\)

Short answer

The lefthand derivative of \(f\) at \(x = 0\) is \(2\). So the correct option is (B) 2.

Step-by-step solution

Identifying the relevant function part

First, identify the function for \(x \leq 0\), which is \(2x+1\).

Taking the derivative

Secondly, the derivative of \(2x+1\) with respect to \(x\) is \(2\). This is because the derivative of \(2x\) is \(2\) and the derivative of 1, a constant, is \(0\). Hence, their sum is \(2 + 0 = 2\).

Evaluating the derivative at point \(x=0\)

Finally, as \(2\) is already a constant, evaluating it at \(x = 0\) does not change it. Therefore, the left-hand derivative of \(f\) at \(x=0\) remains as \(2\).

Key concepts

Piecewise Function

A piecewise function is a type of function that behaves differently depending on the input value. It is defined by multiple sub-functions, each of which applies to a certain interval of the domain. This kind of function is very common in mathematical modeling, where different conditions yield different outcomes.

For the given exercise, the function provided is a piecewise function with two sub-functions. For any input value less than or equal to zero, the function follows one rule (2x + 1), while for any input above zero, it follows another rule (x^2 + 1). When dealing with piecewise functions, it's crucial to consider the domain for each sub-function to correctly apply the rules for derivative calculation.

Exercise Improvement advice: To fully understand piecewise functions, one should practice plotting these functions and identifying their domains and ranges. Moreover, understanding how to apply the function's rules depending on x-values is key in mastering the concept.

Derivative Calculation

Calculating derivatives is a fundamental operation in calculus, which measures the rate at which a function's output value changes as its input value changes. For a linear function like 2x + 1, the derivative is simply the coefficient of x, which in this case is 2. This coefficient represents the slope of the line or the rate of change. For a constant term like 1, the derivative is zero because a constant does not change as the input changes.

Derivative calculation for piecewise functions involves taking the derivative of each piece separately. However, special attention must be paid when calculating derivatives at the point where the function pieces connect, especially if you need to determine continuity or differentiability at that point. Since we’re interested in the left-hand derivative, we focus on the sub-function for x less than or equal to zero, ignoring the sub-function for x greater than zero.

Exercise Improvement advice: When learning derivative calculations, it’s beneficial to practice with different types of functions and understand the rules for each sort (polynomial, trigonometric, exponential, etc.). A strong foundation in the basics makes tackling more complex functions, including piecewise functions, more intuitive.

Calculus Exercise

A calculus exercise, like this one involving a piecewise function and its derivative at a specific point, serves to deepen the student's understanding of concepts like continuity, differentiability, and piecewise-defined functions. For students, it's important not only to obtain the correct answer but also to understand the process of getting there.

In this instance, the exercise is focused on the 'left-hand derivative', a key concept in calculus which refers to the derivative of the function as you approach a certain point from the left side of the graph. The left-hand derivative is often compared with the right-hand derivative to determine a function's differentiability at a point. Left and right-hand derivatives must be equal for a function to be differentiable at that point.

Exercise Improvement advice: To get the most out of calculus exercises, one should always follow the steps methodically: 1) understand the function, 2) calculate its derivatives, and 3) analyze the behavior at the critical points. It’s especially important with piecewise functions to thoroughly check the domains of each sub-function. Repeated practice with step-by-step solutions can be extremely beneficial in building both competence and confidence in solving calculus problems.