Question

Given \(H(0)=17\) and \(H(2)=29,\) approximate \(H^{\prime}(2)\).

Short answer

The approximate value of \(H^{\prime}(2)\) is 6.

Step-by-step solution

Recognize the Formula

To find the derivative at a given point, we can use the formula for the average rate of change between two points as an approximation: \[ H^{\prime}(x) \approx \frac{H(x+\Delta x) - H(x)}{\Delta x} \]where \(\Delta x\) is a small interval. Here, use \(\Delta x = 2\).

Substitute the Values

Substitute the known values into the formula:\[ H^{\prime}(2) \approx \frac{H(2) - H(0)}{2-0} \]Given that \(H(0) = 17\) and \(H(2) = 29\).

Calculate the Expression

Perform the calculation using the substituted values:\[ H^{\prime}(2) \approx \frac{29 - 17}{2} \] Thus,\[ H^{\prime}(2) \approx \frac{12}{2} = 6 \].

Key concepts

Average Rate of Change

The average rate of change is a fundamental concept in calculus used to measure how one quantity changes concerning another over a specified interval. It can be thought of as the mathematical equivalent of the phrase "how fast something is happening." To compute it, we compare two values of a function at different points and evaluate the difference:

• Numerator (change in function value): This is the difference between the function values at two points, say, from point A to point B.
• Denominator (change in input value): This is the difference between the input values at the two corresponding points.

The formula for average rate of change is given by:\[ \frac{f(b) - f(a)}{b-a} \]where \(f\) is your function, and \(a\) and \(b\) are your input values. In practical terms, if you are looking at the speed of a car over time, this average rate of change would represent the average speed over a given time interval. It provides a simple, yet powerful, method to quantify change over a range.

Differential Calculus

Differential calculus is the branch of mathematics focused on rates of change and the slopes of curves. Its main tool: derivatives. A derivative gives us an instantaneous rate of change at a single point, contrasting with the average rate of change over an interval. To understand the derivative better:

• Function slope: The derivative tells how steep a function is at a specific point.
• Instant rate of change: It's like taking a snapshot of the rate of change at a precise instant.
• Tangent line: The derivative at a specific point gives the slope of the tangent line to the function at that point.

When we use derivatives, we're essentially peering into the behavior of a function without needing to look at multiple values. For example, if you're driving, while the speedometer shows your instantaneous speed, the derivative would mathematically confirm that precise speed by analyzing tiny intervals of change around the given moment.

Numerical Methods

Numerical methods provide techniques for approximating solutions to mathematical problems that might be difficult or impossible to solve analytically. In calculus, when exact answers are hard to obtain, numerical methods like derivative approximations come handy. Why are they useful?

• Approximations when exact solutions are unavailable: Sometimes, especially with complex functions, finding an exact derivative or value isn't feasible, so a numerical approximation is necessary.
• Flexibility in application: These methods are flexible and can be adapted to various problems, like estimating integrals, finding roots, or approximating derivatives.
• Ease of calculation: They often transform difficult calculus problems into simpler arithmetic, which can be particularly useful when using a computer or calculator.

In the case of the original problem, numerical methods help estimate the derivative using the average rate of change over a small interval. Even though it’s an approximation, it’s often good enough for practical purposes, especially when dealing with real-world data packed with inherent variability.