Question

Suppose that \(p\) is continuous on \((a, c]\) and differentiable on \((a, c),\) while \(q\) is continuous on \([c, b)\) and differentiable on \((c, b)\). Let $$ f(x)=\left\\{\begin{array}{ll} p(x), & a

Short answer

Question: For a given piecewise function, f, defined as \(f(x) = p(x)\) for \(a < x < c\) and \(f(x) = q(x)\) for \(c < x < b\), prove that the derivative of this function is equal to the derivatives of p(x) and q(x) for the specified intervals and state the conditions for f'(c) to exist. Answer: (a) The derivative of the piecewise function, f, is given by: $$ f^{\prime}(x)=\left\\{\begin{array}{ll} p^{\prime}(x), & a<x<c \\\ q^{\prime}(x), & c<x<b \end{array}\right. $$ (b) The necessary and sufficient conditions for f'(c) to exist are that \(p'(c)\) and \(q'(c)\) exist and are equal: $$ p'(c) = q'(c) $$

Step-by-step solution

Differentiate p(x) and q(x) for the specified intervals

Since \(p(x)\) is differentiable on \((a, c)\), we can find \(p'(x)\) for \(a < x < c\). Similarly, since \(q(x)\) is differentiable on \((c, b)\), we can find \(q'(x)\) for \(c < x < b\).

Define the piecewise derivative function

We will define the piecewise derivative function \(f'(x)\) based on the derivatives of \(p(x)\) and \(q(x)\): $$ f^{\prime}(x)=\left\\{\begin{array}{ll} p^{\prime}(x), & a<x<c \\\ q^{\prime}(x), & c<x<b \end{array}\right. $$

Prove the existence of f'(c) under additional conditions

The derivative of f at the point c exists if and only if the following conditions are met: 1. The left-hand derivative of f at point c exists and is equal to \(p'(c)\). 2. The right-hand derivative of f at point c exists and is equal to \(q'(c)\). 3. \(p'(c) = q'(c)\). The necessary and sufficient conditions for these are that \(p'(c)\) and \(q'(c)\) exist and are equal: $$ p'(c) = q'(c) $$ Under these conditions, the derivative of f at the point c exists: $$ f'(c) = p'(c) = q'(c) $$ Thus, our final answer is: (a) $$ f^{\prime}(x)=\left\\{\begin{array}{ll} p^{\prime}(x), & a<x<c \\\ q^{\prime}(x), & c<x<b \end{array}\right. $$ (b) The additional conditions are that \(p'(c)\) and \(q'(c)\) exist and are equal: $$ p'(c) = q'(c) $$

Key concepts

Understanding Continuity

Continuity is a fundamental concept in calculus. A function is continuous over an interval if you can draw its graph without lifting your pencil. For the function \( f(x) \) defined as piecewise with \( p(x) \) and \( q(x) \), continuity implies that in their respective domains, \( p(x) \) and \( q(x) \) have no jumps or breaks.
To ensure continuity for a piecewise function at a specific point \( c \), the left-hand limit (as \( x \) approaches \( c \) from the left) and the right-hand limit (as \( x \) approaches \( c \) from the right) must be equal. This also means the actual value of the function at \( c \) (if defined) equals these limits.
For example:

• \( \lim_{{x \to c^-}} p(x) = \lim_{{x \to c^+}} q(x) \)
• If \( p(c) \) and \( q(c) \) are equal, then the function \( f(x) \) is continuous at \( c \).

Differentiation Essentials

Differentiation involves finding the derivative of a function, representing its rate of change. For piecewise functions, differentiate each piece separately where it is defined. Here, \( p(x) \) is differentiable on \((a, c)\), and \( q(x) \) is differentiable on \((c, b)\). Therefore, these derivatives form the piecewise derivative function \( f'(x) \).
Calculating Piecewise Derivatives:

• For \( a < x < c \), \( f'(x) = p'(x) \).
• For \( c < x < b \), \( f'(x) = q'(x) \).

Piecewise derivatives are computed separately for each interval and highlight how the function changes in its different sections.

Conditions for Existence of Derivative

For the derivative at the point \( c \) to exist, both the left and right derivatives at \( c \) must exist and be equal. This ensures the smoothness of the transition between \( p(x) \) and \( q(x) \) at \( c \).
Necessary conditions:

• The left-hand derivative exists: \( p'(c) \).
• The right-hand derivative exists: \( q'(c) \).
• Equal left and right derivatives: \( p'(c) = q'(c) \).

If these conditions are satisfied, the function is differentiable at \( c \), meaning \( f'(c) = p'(c) = q'(c) \). This is critical for piecewise functions, as it ensures they are not only continuous but also smooth at the transition point.