Problem 53 - Calculus Early Transcendentals

Question

Estimate \(f(3.85)\) given that \(f(4)=3\) and \(f^{\prime}(4)=2\).

Step-by-Step Solution

Answer

Question: Estimate the value of the function \(f(x)\) at \(x=3.85\) if \(f(4)=3\) and \(f^{\prime}(4)=2\). Answer: The estimated value of \(f(3.85)\) is approximately \(2.7\).

Step 1: Formula for Tangent Line Approximation

To estimate the function value, we can use the tangent line approximation, which is given by the formula: $$f(x) \approx f(a) + f^{\prime}(a)(x-a)$$, where \(a\) is a point close to \(x\), and in our case, \(a=4\) and \(x=3.85\).

Step 2: Plug in Given Values

We are given \(f(4)=3\) and \(f^{\prime(}(4)=2\). Now, we can plug these values into the tangent line approximation formula: $$f(3.85) \approx f(4) + f^{\prime}(4)(3.85-4)$$ $$f(3.85) \approx 3 + 2(3.85-4)$$

Step 3: Compute the Approximation

Now, we can calculate the result: $$f(3.85) \approx 3 + 2(-0.15)$$ $$f(3.85) \approx 3 - 0.3$$ $$f(3.85) \approx 2.7$$ So, the estimate for \(f(3.85)\) is \(2.7\).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, providing a way to measure how a function changes as its input changes. The derivative of a function at a point tells us the slope of the tangent line to the curve at that point. This means it's essentially measuring the rate of change or the velocity of a function.

• If you think about a car's speedometer, the reading is the derivative of the car's position function over time.
• For a function, it describes how quickly or slowly the function's value is increasing or decreasing as you move along the graph.

In the context of the tangent line approximation formula used in our problem, the derivative at a specific point gives us the slope of the line that we use to approximate the function's value at nearby points. For our exercise, knowing that \(f'(4) = 2\) provided essential information about how the function \(f\) behaves around \(x = 4\).

Function Estimation Through Tangent Line Approximation

Function estimation is a useful tool when you cannot calculate the exact value of a function at a certain point, but you have enough information to approximate it. In our exercise, we used the tangent line approximation method to estimate \(f(3.85)\). This method relies on knowledge of the function value and its derivative at a nearby point.

• The approximation is calculated using the tangent line to the function at a known point \((a, f(a))\) with \(a = 4\) in our question.
• The formula \(f(x) \approx f(a) + f'(a)(x-a)\) helps us estimate the value of the function \(f\) at a point close to \(a\).
• By plugging in the values \(f(4)=3\) and \(f'(4)=2\), we determine that: \[ f(3.85) \approx 3 + 2(-0.15) = 2.7 \]

This estimation is effective as long as \(x\) is sufficiently close to \(a\), ensuring that the curve doesn’t deviate significantly from the tangent line.

Solving Calculus Problems with Tangent Lines

Tangent line approximation is a practical technique in solving calculus problems, especially when dealing with real-world situations where exact values are difficult to calculate. In calculus, such approaches help simplify complex functions and provide intuitive ways to analyze changes without precise measurements.

• This method is particularly helpful in situations where you need quick calculations, such as in physics or engineering.
• Understanding how to manipulate the derivative and apply it to find function estimations is a foundational skill in mathematics.
• Practicing with different problems improves your ability to quickly and accurately use approximation methods.

In learning calculus problem-solving strategies, the tangent line approximation serves as an introduction to more advanced concepts like differentials and linearization, bridging the gap between pure math and its practical applications. Engaging with exercises like these strengthens your problem-solving skills and prepares you for tackling more complex mathematical challenges.