Question

Simplify the difference quotient\(\frac{f(x+h)-f(x)}{h}\) for the following functions. $$f(x)=4 x-3$$

Short answer

Answer: The simplified difference quotient is \(4\).

Step-by-step solution

Write down the difference quotient formula

The difference quotient formula is given by: $$\frac{f(x+h)-f(x)}{h}$$

Substitute the given function into the difference quotient formula

The given function is \(f(x) = 4x - 3\). We need to find \(f(x+h)\) and \(f(x)\). \(f(x) = 4x - 3\) \(f(x+h) = 4(x+h) - 3\) Now substitute these expressions into the difference quotient formula: $$\frac{(4(x+h)-3) - (4x-3)}{h}$$

Simplify the expression

Let's simplify the expression in the numerator: $$\frac{4x+4h-3-4x+3}{h}$$ Observe that \(4x\) and \(-4x\) cancel out, and \(-3\) and \(+3\) cancel out. So, we are left with: $$\frac{4h}{h}$$

Simplify the expression further by canceling out h

Since we have an \(h\) term in both the numerator and the denominator, we can cancel them out: $$\frac{4h}{h} = 4$$ So the simplified difference quotient of the function \(f(x) = 4x - 3\) is \(4\).

Key concepts

Simplifying Difference Quotient

The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval. To simplify the difference quotient, \( \frac{f(x+h)-f(x)}{h} \), we start by evaluating the function at \( x+h \) and \( x \) separately, and then we find the difference between these two values.

In our specific exercise, we have the function \( f(x) = 4x - 3 \). Following algebraic operations, we substitute \( x+h \) and \( x \) into the function to obtain \( f(x+h) = 4(x+h) - 3 \) and \( f(x) = 4x - 3 \), respectively. Then, we place these into our difference quotient formula. Through simplification, we cancel out like terms and factors to reach a simplified expression, which in this case is \( 4 \).

The key to simplifying the difference quotient effectively lies in a systematic approach: expanding the expressions, combining like terms, and reducing fractions. For visual learners, it may help to draw each step or use different colors to identify common terms that can be eliminated through the simplification process.

Calculus

Calculus is an advanced branch of mathematics focused on change and motion. Within its study, the difference quotient plays a vital role as it leads to the derivative. A derivative can be understood as the instantaneous rate of change, and it is found by taking the limit of the difference quotient as \( h \rightarrow 0 \).

Simplification of expressions, such as the difference quotient, is a crucial skill within calculus. By mastering this, students can better understand how functions behave at specific points and how they are transformed when changes are applied to their input variables. In the exercise we considered, the simplified difference quotient for \( f(x) = 4x - 3 \) effectively tells us that for this linear function, the rate of change is constant, and this rate of change is the derivative \( f'(x) \).

Algebraic Simplification

Algebraic simplification is the process of making a mathematical expression more manageable and understandable by reducing it to its simplest form. This often involves combining like terms, factoring, expanding expressions, and canceling common factors.

In our exercise, we utilized several algebraic simplification techniques to transform the difference quotient. Initial steps involved distributing the multiplier over a sum, combining terms, and then canceling out common factors (in this case, the variable \( h \)). A strong foundation in algebra is critical to success in calculus, as it enables students to manipulate and simplify expressions before applying more complex calculus concepts.

To assist students further, educators often suggest practicing similar difference quotient problems with varying functions to reinforce the connection between algebraic simplification and calculus concepts. This repetitive practice ensures that the seemingly mechanical process of simplifying expressions becomes intuitive and second nature.