Problem 50 - Single Variable Calculus

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. $$ \begin{array}{l} \text { If } h=g \circ f, \text { where } g \text { and } f \text { are differentiable everywhere, }\\\ \text { then } h(x+\Delta x) \text { \& } g(f(x))+g^{\prime}(f(x)) f^{\prime}(x) \Delta x \text { . } \end{array} $$

Step-by-Step Solution

Answer

The given statement is true. The approximation \(h(x+\Delta x) \approx g(f(x))+g^{\prime}(f(x)) f^{\prime}(x) \Delta x\) is valid for any differentiable functions \(g\) and \(f\). This is based on the chain rule for derivatives and linear approximation, which is applicable to any differentiable functions.

Step 1: Write h(x) as the composition of g and f

As per the given statement, we have \(h(x) = g(f(x))\). Now, let's find \(h'(x)\) using the chain rule. #Step 2: Apply the chain rule to h#

Step 2: Apply the chain rule to h

By the chain rule, we have: \[h'(x) = g'(f(x)) \cdot f'(x)\] Now let's estimate \(h(x + \Delta x)\) using this derivative. #Step 3: Estimate h(x + Δx)#

Step 3: Estimate h(x + Δx)

Using the linear approximation, we have: \[h(x + \Delta x) \approx h(x) + h'(x) \cdot \Delta x\]. Since \(h(x)=g(f(x))\) and \(h'(x)=g'(f(x))\cdot f'(x) \), we get: \[h(x+\Delta x)\approx g(f(x))+g'(f(x))\cdot f'(x) \Delta x\] In conclusion: The given statement is true. The approximation \(h(x+\Delta x) \approx g(f(x))+g^{\prime}(f(x)) f^{\prime}(x) \Delta x\) is valid for any differentiable functions \(g\) and \(f\). This is based on the chain rule for derivatives and linear approximation, which is applicable to any differentiable functions.

Key Concepts

Composition of Functions

In mathematics, the composition of functions is an operation that takes two functions and combines them to form a new function. Imagine you have two functions, \( f(x) \) and \( g(x) \). When you perform a composition, you essentially put one function inside the other. The notation \( g \circ f \) means "\( g \) of \( f \)." In this scenario, you first apply \( f \) to \( x \), and then apply \( g \) to the result of \( f(x) \). This results in the new function \( h(x) = g(f(x)) \).

When working with compositions, understanding the order of operations is crucial. Always start with the inner function and work your way out.

• The composition \( g \circ f \) means apply \( f \) first, then \( g \)
• This operation only makes sense if the output of \( f \) matches the input required by \( g \)

Composition of functions is a powerful tool in calculus, as it allows for the simplification and manipulation of complex problems.

Differentiable Functions

A function is said to be differentiable at a point if it has a derivative there, which means it can be approximated by a linear function at that point. Differentiability is an extension of continuity; if a function is differentiable at a point, it is also continuous there. In our exercise, both \( g \) and \( f \) are assumed to be differentiable everywhere, implying they have smooth and continuous behavior for all real numbers.

Differentiable functions have some key properties that make them invaluable in calculus:

• They allow us to find the slope of the tangent line at any point on the graph of the function.
• The derivative provides a rate of change, demonstrating how changes in the input affect the output.

When two functions are composed to form \( h(x) = g(f(x)) \), the differentiability of \( h \) depends on the differentiability of both \( g \) and \( f \). Utilizing the chain rule, we can effectively find the derivative of such composite functions, which is essential for solving real-world problems.

Linear Approximation

Linear approximation is a method used to estimate values of a function near a given point using a tangent line. It's grounded in the idea that, near a specific point, the function can be well-approximated by its tangent line. For the function \( h(x) = g(f(x)) \), if we know \( h'(x) \), then we can use the formula \( h(x + \Delta x) \approx h(x) + h'(x) \cdot \Delta x \), giving us a linear approximation of how \( h(x) \) behaves around \( x + \Delta x \).

Key elements of linear approximation include:

• It's highly accurate for very small changes in \( x \) (\( \Delta x \) is small).
• It simplifies complex calculations by reducing them to a straight-line estimate.
• This approach forms the basis of the linearization concept in calculus.

Such approximations help us understand the local behavior of functions, which is particularly valuable when dealing with composite functions like \( g(f(x)) \). In practice, this means that small changes in \( x \) lead to close estimates of function values without needing to calculate \( g(f(x + \Delta x)) \) directly.