Problem 56 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose \(h=f \circ g\). Find \(h^{\prime}(0)\) given that \(f(0)=6\), \(f^{\prime}(5)=-2, g(0)=5\), and \(g^{\prime}(0)=3\)

Step-by-Step Solution

Answer

Using the Chain Rule and the given information, we can find \(h'(0)\) as follows: \[ h'(0) = (f \circ g)'(0) = f'(g(0)) \cdot g'(0) = f'(5) \cdot 3 = -2 \cdot 3 = -6.\] Thus, \(h'(0) = -6\).

Step 1: Write down the Chain Rule

The Chain Rule is used when finding the derivative of a composite function, and it states: \[ (f \circ g)'(x) = f'(g(x)) \cdot g'(x).\] In our problem, we want to find \(h'(0)\) or \((f \circ g)'(0).\)

Step 2: Evaluate \((f \circ g)'(0)\) using the given information

Using the Chain Rule and the information provided in the problem, let's find \((f \circ g)'(0)\): \( (f \circ g)'(0) = f'(g(0)) \cdot g'(0) \) We are given that \(g(0) = 5\) and \(g'(0) = 3\), as well as \(f'(5) = -2\). We substitute these values into the equation: \( (f \circ g)'(0) = f'(5) \cdot 3 \)

Step 3: Complete the calculation for \(h'(0)\)

Now that we have substituted the given values, we can find \(h'(0)\) as follows: \( h'(0) = -2 \cdot 3 = -6 \) The derivative of \(h\) at \(x=0\) is \(-6\).

Key Concepts

Composite Functions

Understanding composite functions is essential when working with complex relationships between variables. A composite function is created when one function is applied to the result of another function. Mathematically, if you have two functions, say, f and g, the composite function h, which is written as h = f \( \circ \) g, means you first apply g to an input x, and then apply f to the result.

For example, if g(x) = x + 1 and f(x) = 2x, then the composite function h(x) = f(g(x)) would give h(x) = 2(x + 1). Simply put, you first execute the function g, which adds 1, and then execute the function f, which multiplies the result by 2. It's crucial in calculus to know how to combine and work with these composite functions, especially when dealing with their derivatives.

Derivative Calculation

The derivative of a function measures how the function's output changes as its input changes. In calculus, derivative calculation embodies the fundamental process of finding the rate at which a quantity changes. For functions that are combinations of numerous simpler functions, such as composite functions, this becomes slightly tricky, and this is where the chain rule comes into play.

When you're faced with the task of computing the derivative of a composite function, like h=f \( \circ \) g, the chain rule provides an effective method. This rule tells us that the derivative of the composite function h can be found by multiplying the derivative of f at g(x) by the derivative of g at x. In mathematical terms, this is expressed as h' = f'(g(x)) \cdot g'(x). Using this rule simplifies what would otherwise be a very complicated derivative calculation.

Function Derivative

Triggering the change in a function's value with respect to a change in the input is what a function derivative represents. It paints a picture of the rate of change or the slope of the function at any point on its curve. The derivative itself is a function that can be evaluated at given points to understand the behavior of the original function at those points.

Bringing this into the realm of composite functions—when we want the derivative of a composite function at a certain point, we are looking for the instantaneous rate of change of one function after it has been modified by another. As seen in our exercise, the derivative of h at x=0, written as h'(0), tells us how quickly h is changing right at the moment when x is zero. The way we calculate this using the chain rule is a powerful demonstration of how interconnected functions react to changes and it's a crucial concept for anyone delving into calculus.