Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

Given that \(f(0)=1\) and \(f^{\prime}(0)=2\), find \(g^{\prime}(0)\) where \(g(x)=\cos f(x)\)

Step-by-Step Solution

Answer

\(-2 \sin(1)\)

Step 1: Identify Given Information

We are given that \(f(0) = 1\) and \(f'(0) = 2\). This means that the function \(f\) evaluated at \(x = 0\) is equal to 1, and its derivative at this point is equal to 2.

Step 2: Define the Function to Differentiate

We need to find \(g'(0)\). The function \(g(x)\) is given by \(g(x) = \cos(f(x))\), which is the composition of the cosine function and \(f(x)\).

Step 3: Apply Chain Rule

To find the derivative \(g'(x)\), we apply the chain rule. The chain rule states that if \(g(x) = \,\cos(f(x))\), then \(g'(x) = -\sin(f(x)) \cdot f'(x)\).

Step 4: Evaluate the Derivative at 0

Now, substitute \(x = 0\) into \(g'(x)\). Using the chain rule result from Step 3, we have: \[ g'(0) = -\sin(f(0)) \cdot f'(0) \]Substitute the given values \(f(0) = 1\) and \(f'(0) = 2\):\[ g'(0) = -\sin(1) \cdot 2 \]Simplifying, we obtain:\[ g'(0) = -2\sin(1) \]

Step 5: Present the Final Result

The value of \(g'(0)\) is \(-2\sin(1)\). This is the derivative of the function \(g(x) = \cos(f(x))\) at \(x = 0\), taking into account the given information about \(f(x)\).

Key Concepts

Derivative

Derivatives play a fundamental role in calculus, acting as a tool to measure how a function changes as its input changes. In simple terms, the derivative tells you the "slope" or rate of change of a function at any point. When given a function \( f(x) \), the derivative \( f'(x) \) represents how \( f \) changes as \( x \) changes.

Imagine you're driving a car; the speedometer tells you the speed at that exact moment. Similarly, the derivative at a point tells you how fast the function's value is changing at that point. Understanding derivatives is essential, not just for solving practical problems like finding the slope of a curve, but also for more advanced work in mathematics and science. It's the foundation for analyzing trends and behaviors in mathematical functions.

• It provides the average rate of change between two points.
• It's useful for finding maximum and minimum values of functions (critical points).

Function Composition

Function composition involves the operation of applying one function to the results of another. If you have two functions, say \( f(x) \) and \( g(x) \), the composition \( (g \circ f)(x) \) means you first apply \( f \) to \( x \) and then apply \( g \) to the result of \( f \).

In the given exercise, the function \( g(x) = \cos(f(x)) \) is a composed function where \( \cos \) is applied after the function \( f(x) \). This composition makes finding the derivative more complex, which is where the chain rule becomes necessary.

• It allows more complex functions to be broken down into simpler parts.
• It often requires using the chain rule for differentiation.

Function composition is a powerful concept, especially useful in creating more complex models from simple functions. It enables smoother calculations when managing multiple operations simultaneously.

Trigonometric Functions

Trigonometric functions like \( \sin \), \( \cos \), and \( \tan \) describe the relationships between the angles and sides of a triangle. In calculus, these functions are also fundamental because they help model periodic phenomena such as sound waves or tides.

In the original exercise, \( g(x) = \cos(f(x)) \), the cosine function is applied to the outputs of another function \( f(x) \). Cosine is one of the primary trigonometric functions, and it shows up in many contexts beyond triangles, especially in calculus to handle oscillating phenomena.

• Cosine returns values between -1 and 1, representing wave-based or circular phenomena.
• Its derivative is \(-\sin(x)\), which plays a crucial role when combined with other derivatives, particularly in chain rule applications.

Understanding these functions within calculus allows for the modeling of more complex mathematical relationships and helps solve real-world problems involving waves or periodic motion.