Problem 47 - Calculus - AP Edition

Question

Show that $$f(x)=\left\\{\begin{array}{ll} x^{2}+1, & x \leq 1 \\ 2 x, & x>1 \end{array}\right.$$ is continuous and differentiable at \(x=1 .\) Sketch the graph of \(f\)

Step-by-Step Solution

Answer

The function is continuous and differentiable at \(x=1\). Graph: parabola \(x^2+1\) for \(x \leq 1\), line \(2x\) for \(x>1\) with smooth transition.

Step 1: Check continuity at x=1

For a function to be continuous at a point, the limit from the left must equal the limit from the right and the value of the function at that point. First, evaluate the limit from the left: \[\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^-}} (x^2 + 1) = 1^2 + 1 = 2.\] Next, evaluate the limit from the right: \[\lim_{{x \to 1^+}} f(x) = \lim_{{x \to 1^+}} (2x) = 2 \times 1 = 2.\] Finally, evaluate \(f(1)\): \[f(1) = 1^2 + 1 = 2.\] Since all three values are equal, the function is continuous at \(x = 1\).

Step 2: Check differentiability at x=1

A function is differentiable at a point if the derivative from the left equals the derivative from the right. Calculate the derivative for \(x \leq 1\): \[f'(x) = 2x, \text{ so from the left, } f'(1^-) = 2 \times 1 = 2.\] Calculate the derivative for \(x > 1\): \[f'(x) = 2, \text{ so from the right, } f'(1^+) = 2.\] Since both derivatives are equal, the function is differentiable at \(x = 1\).

Step 3: Sketch the graph of f

To sketch the graph, plot the two pieces of the function. For \(x \leq 1\), plot \(f(x) = x^2 + 1\), a parabola opening upwards, starting at 2 when \(x = 1\). For \(x > 1\), plot \(f(x) = 2x\) as a straight line starting at the point \((1, 2)\). The graph should show a smooth transition at \(x=1\), reflecting both continuity and differentiability.

Key Concepts

Continuity

Continuity is a fundamental concept in calculus. It means that at any given point of the function, you can draw the graph without lifting your pencil. Specifically, a function is continuous at a point if the limit as it approaches that point from both the left and the right equals the value of the function at that point. In mathematical terms, for a function \( f \) to be continuous at \( x = c \), it must satisfy:

• \( \lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = f(c) \)

Applying this to the function \( f(x) \) given in the exercise:

• \( \lim_{{x \to 1^-}} (x^2 + 1) = 2 \)
• \( \lim_{{x \to 1^+}} (2x) = 2 \)
• \( f(1) = 2 \)

Since all these values match, \( f(x) \) is continuous at \( x = 1 \). The continuity ensures that there is a smooth, unbroken curve on the graph at this point.

Differentiability

Differentiability involves the smoothness of a function at a point. If a function is differentiable at a point, it also means it is continuous there, but the converse is not always true. Differentiability implies that the function has a defined tangent line at that point, meaning the function does not have sharp turns or cusps.
For \( f(x) \) to be differentiable at \( x = c \), the left-hand derivative and the right-hand derivative at \( c \) must be equal:

• Left derivative: \( \lim_{{h \to 0^-}} \frac{f(c + h) - f(c)}{h} \)
• Right derivative: \( \lim_{{h \to 0^+}} \frac{f(c + h) - f(c)}{h} \)

For the provided function:

• \( f'(1^-) = 2 \)
• \( f'(1^+) = 2 \)

Since both derivatives are equal, \( f \) is differentiable at \( x = 1 \). This differentiability at \( x = 1 \) ensures that the function has no breaks or sharp corners in its graph at that point.

Piecewise Function

A piecewise function is a type of function built from multiple sub-functions, each applying to a certain interval of the main function's domain. It allows us to sculpt functions that behave differently across various intervals. This flexibility makes them essential in modeling real-world scenarios where rules change based on circumstances.
The function given in the exercise is a piecewise function:

• For \( x \leq 1 \), \( f(x) = x^2 + 1 \)
• For \( x > 1 \), \( f(x) = 2x \)

When analyzing a piecewise function, special care is needed at the points where the function's formula changes, known as the "junction points." To determine continuity and differentiability at these points, check limits and derivatives as done in the exercise.
These junction points often demand extra attention in problem-solving, ensuring all pieces fit smoothly together without gaps or jumps, which is demonstrated by the continuous and differentiable nature of the function at \( x = 1 \). Understanding how these pieces interact helps in sketching a complete and accurate graph.