Problem 56 - Calculus Single Variable

Question

Use the given information to estimate \(f^{\prime}(c)\) at the given point \(c\) $$ f(4)=5.7 \text { and } f(4.1)=6.2, c=4.1 $$

Step-by-Step Solution

Answer

The estimated value of \( f'(c) \) at \( c = 4.1 \) is 5.

Step 1: Identify the Difference Quotient Formula

The derivative of a function at a point can be approximated as the difference quotient: \( f'(c) \approx \frac{f(x+h) - f(x)}{h} \), where \( h \) is a small increment.

Step 2: Assign Values to the Formula

Here, we are given \( f(4) = 5.7 \) and \( f(4.1) = 6.2 \). We want to estimate \( f'(4.1) \). Assign \( x = 4 \) and \( h = 0.1 \), so the formula becomes \( f'(4.1) \approx \frac{f(4.1) - f(4)}{0.1} \).

Step 3: Plug in the Values

Substitute the given values: \( f'(4.1) \approx \frac{6.2 - 5.7}{0.1} \).

Step 4: Calculate the Result

Compute the numerator: \( 6.2 - 5.7 = 0.5 \). Then divide by \( 0.1 \): \( f'(4.1) \approx \frac{0.5}{0.1} = 5 \).

Key Concepts

Difference Quotient

In differential calculus, the concept of a difference quotient is pivotal for understanding how derivatives work. A difference quotient helps us approximate the rate of change of a function over a small interval. Imagine you are examining how a car's speed changes over a very short distance; the difference quotient gives us that insight by measuring changes between two points.

• It is expressed as: \( \frac{f(x+h) - f(x)}{h} \), where \( f(x) \) represents the function value at a point \( x \), and \( h \) is a very small increment.
• This approach helps us "zoom in" on the curve of a function, by looking at the slope between two closely spaced points.
• As \( h \) approaches zero, the difference quotient becomes the derivative, represented by \( f'(x) \).

In the given exercise, understanding the difference quotient allows us to estimate \( f'(c) \) at the point \( c = 4.1 \), by looking at how the function values change from \( 4 \) to \( 4.1 \).

Derivative Estimation

Estimating a derivative involves calculating the rate at which the function's value changes at a particular point. This is important because direct measurement of the derivative can be difficult or impossible in many real-world applications. We rely on approximation techniques like using a difference quotient.

• In the context of the given exercise, we do not have \( f'(x) \) directly. Instead, we approximate it using known function values.
• By substituting \( f(4.1) \) and \( f(4) \) into the difference quotient formula, we derive an estimate for \( f'(4.1) \).
• This is achieved by calculating \( \frac{6.2 - 5.7}{0.1} \), which gives us the estimated derivative.

By understanding derivative estimation, we have the tools to analyze complex systems where precise data points are scarce, yet change over tiny intervals is immensely informative.

Function Approximation

Function approximation is a crucial technique when dealing with continuously changing systems. It allows us to model and predict behaviors by looking at a small section of the function and expanding our understanding from there.

• In practical terms, this involves estimating or "guessing" the behavior of a function at points where direct measurement isn't feasible.
• In the exercise, although we don't know \( f'(4.1) \) exactly, we can approximate it by evaluating the slope between two close function points: \( f(4) \) and \( f(4.1) \).
• Approximations are especially useful in scientific and engineering contexts, where complex models must be simplified to predict outcomes efficiently.

Grasping function approximation empowers us to apply mathematical concepts to everyday life situations, predicting future trends and examining past patterns with incomplete data.