Problem 19 - Calculus for Biology and Medicine

Question

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=\sqrt{x} ; c=4 $$

Step-by-Step Solution

Answer

\(f(c+h) - f(c) = \sqrt{4+h} - 2\)

Step 1: Substitute the Function

The exercise requires us to compute the difference \(f(c+h)-f(c)\). We know that \(f(x) = \sqrt{x}\). Begin by substituting \(f(c+h)\) and \(f(c)\) as \(f(x)\) indicates.So:\[f(c+h) = \sqrt{c+h}\quad\text{and}\quad f(c) = \sqrt{c}\]

Step 2: Apply the Specific Point

Now, apply the given value \(c = 4\) into the expressions we found in Step 1.\[f(c+h) = \sqrt{4+h}\quad\text{and}\quad f(c) = \sqrt{4}\]

Step 3: Calculate the Difference

Subtract \(f(c)\) from \(f(c+h)\) as indicated in the problem statement.\[f(c+h) - f(c) = \sqrt{4+h} - \sqrt{4}\]Since \(\sqrt{4}\) is a known value:\[\sqrt{4} = 2\]So the expression becomes:\[f(c+h) - f(c) = \sqrt{4+h} - 2\]

Key Concepts

Function Difference

When working with functions, understanding the concept of a function difference is crucial. A function difference indicates how much a function's value changes between two points. In mathematical terms, this is often expressed as \( f(c+h) - f(c) \), where \( f(x) \) is the function in question.

• The function difference helps identify trends and changes in the function's behavior over an interval.
• In our exercise, we see this as the contrast between \( \sqrt{4+h} \) and \( \sqrt{4} \).

The objective is to analyze these changes to understand the function better, which is pivotal in calculus, especially in finding derivatives and understanding limits.

Square Root Function

The square root function, represented as \( f(x) = \sqrt{x} \), is a fundamental concept in mathematics.

• This function involves finding a number which, when multiplied by itself, yields the original number, \( x \).
• For instance, in our exercise, \( \sqrt{4} = 2 \), because \( 2 \times 2 = 4 \).

Square root functions are inherently non-linear, meaning they do not form a straight line when graphed. Instead, they curve, with the curve getting shallower as \( x \) increases. Understanding square root functions is key for solving problems that involve roots, radicals, and understanding growth patterns in natural phenomena.

Algebraic Manipulation

Algebraic manipulation is a technique used to simplify mathematical expressions and solve equations. It is essential when working with function differences and other calculus-related problems.

• The goal is to transform an expression into a simpler or more workable form.
• In our problem, algebraic manipulation helps simplify \( \sqrt{4+h} - 2 \).

Performing algebraic manipulation often involves applying arithmetic operations, factorization, and understanding mathematical properties. This skill is crucial in making complex problems much more manageable, allowing us to isolate certain variables or expressions to better analyze their components.