Problem 70 - Calculus an Applied Approach

Question

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=3-g(x) $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)\) at the point \(x=2\), represented as \(f^{\prime}(2)\), is 2.

Step 1: Find the derivative of function \(f(x)\)

The derivative of function \(f(x)\) can be found using the rule of differentiation: \(f^{\prime}(x)=0-g^{\prime}(x)\). Here, the derivative of the constant 3 is 0, and the derivative of \(g(x)\) should be represented by \(g^{\prime}(x)\). Therefore, \(f^{\prime}(x)=-g^{\prime}(x)\).

Step 2: Calculating \(f^{\prime}(2)\)

To find \(f^{\prime}(2)\), substitute \(x=2\) into the previously found expression for \(f^{\prime}(x)\), i.e. \(f^{\prime}(2)=-g^{\prime}(2)\). Given that \(g^{\prime}(2)=-2\), substituting it into the equation will give: \(f^{\prime}(2)=-(-2)\).

Step 3: Finalizing the Solution

Finally, simplifying \(f^{\prime}(2)=-(-2)\), we get: \(f^{\prime}(2)=2\). So, the derivative of function \(f(x)\) at the point \(x=2\) is 2. The \(h(x)\) function wasn't used as it wasn't present in the given function \(f(x)\).

Key Concepts

Differentiation Rules

Differentiation is a crucial process in calculus used to determine the rate of change of a function with respect to one of its variables. It allows us to find the derivative of a function, which can be interpreted as the slope of the tangent to the graph of the function at any point.

To differentiate a function, we often utilize specific differentiation rules. Two of the most common rules are the constant rule and the subtraction rule. The constant rule states that the derivative of any constant is zero. For example, if we have a function like \(f(x) = 3\), then its derivative is \(f'(x) = 0\).

• Constant Rule: \(\frac{d}{dx}[c] = 0\)
• Subtraction Rule: \(\frac{d}{dx}[u - v] = u' - v'\) where \(u'\) and \(v'\) are derivatives of \(u\) and \(v\) respectively.

In the given exercise, the function was \(f(x) = 3 - g(x)\). By applying the constant rule and subtraction rule, we found \(f'(x) = -g'(x)\). Understanding these basic rules simplifies differentiation tasks significantly and provides a foundation for tackling more complex problems.

Applications of Derivatives

Derivatives have numerous applications, both mathematical and real-world. They are extensively used to find the rate of change of a quantity. This property makes them valuable in physics, engineering, economics, and biology.

• In physics, derivatives help determine velocities and accelerations of moving objects.
• In economics, they are used to calculate marginal cost and revenue.
• Engineers use them to assess the sensitivity of certain variables in designs.

For our specific function \(f(x) = 3 - g(x)\), finding the derivative \(f'(x)\), and then evaluating it at a particular point \(x = 2\), allowed us to understand how \(f(x)\) is changing at that point.

By substituting \(x = 2\) into our derivative, we saw that the rate of change of \(f(x)\) at this point is 2. This kind of evaluation is critical when analyzing functions to make predictions or to understand behavior at a given moment. For instance, in business, such analysis can determine optimal points for maximizing profit or minimizing cost.

Function Analysis

Function analysis involves understanding the behavior, characteristics, and properties of functions. It requires evaluating a function at various points or over an interval to assess how the function behaves.

In our exercise, the function analysis focused on \(f(x)\) by looking at its derivative. The derivative tells us interesting details about the function's graph, such as whether it's increasing or decreasing at a specific point.

• If \(f'(x) > 0\), the function is increasing at that point.
• If \(f'(x) < 0\), the function is decreasing at that point.
• If \(f'(x) = 0\), the function may have a local maximum or minimum.

For \(f'(2) = 2\), since 2 is positive, it informs us that \(f(x)\) is increasing at \(x = 2\).

Function analysis can further involve evaluating critical points, observing concavity, and finding points of inflection, paving the way to understanding the entire picture of a function's behavior. Thus, derivative calculation and analysis provide significant insights, vital for comprehensive function behavior understanding.