Problem 57
Question
find and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ for the given function. $$ f(x)=3 x+7 $$
Step-by-Step Solution
Verified Answer
The simplified form of the difference quotient for the function \(f(x)=3x+7\) is 3.
1Step 1: Substitute the function into the difference quotient
Firstly, substitute the function \(f(x)=3x+7\) into the difference quotient. The difference quotient is given by \(\frac{f(x+h)-f(x)}{h}\). So, upon substituting the function into the former, it will result in \(\frac{3(x+h)+7-(3x+7)}{h}\).
2Step 2: Simplify the difference quotient
Now simplify the difference quotient to reduce it to a smallest form. The expression inside the numerator simplifies to \(3h\). Therefore, the difference quotient becomes \(\frac{3h}{h}\).
3Step 3: Cancel out the common terms
Lastly, on simplifying the equation further, the \(h\) in the numerator and denominator cancels out leaving just the number 3 as the final result.
Key Concepts
AlgebraSimplifying Algebraic ExpressionsLimit Concepts
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a language of its own that allows us to formulate relationships and solve problems involving unknown quantities, referred to as variables. In the context of our difference quotient exercise, we work with the variable 'x' and a change in 'x', which is denoted as 'h'.
When engaging with algebraic expressions, particularly in calculus, students are often introduced to the concept of the difference quotient. This concept is pivotal in defining the derivative of a function—a fundamental cornerstone in calculus. Algebra provides the tools necessary to rearrange, simplify, and manipulate such expressions to reveal underlying behaviors of functions, like the rate of change.
Understanding algebra is crucial for solving the difference quotient because you first need to recognize and apply the conventional rules of algebra to combine like terms and simplify expressions.
When engaging with algebraic expressions, particularly in calculus, students are often introduced to the concept of the difference quotient. This concept is pivotal in defining the derivative of a function—a fundamental cornerstone in calculus. Algebra provides the tools necessary to rearrange, simplify, and manipulate such expressions to reveal underlying behaviors of functions, like the rate of change.
Understanding algebra is crucial for solving the difference quotient because you first need to recognize and apply the conventional rules of algebra to combine like terms and simplify expressions.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is an essential skill in algebra, serving as the foundation for more complex mathematical concepts. Simplification makes equations and expressions easier to work with and understand. It often involves combining like terms, expanding expressions, and canceling out common factors in a fraction.
For example, in our exercise, we start with a complex fraction and simplify it step by step. Initially, by expanding and subtracting terms within the numerator, and finally, by canceling out common factors in both the numerator and the denominator. In this case, the 'h' terms cancel each other out, leaving a constant as the simplified result of the difference quotient.
In practice, simplifying an algebraic expression is the right approach to prep the terrain for further mathematical procedures, such as taking limits or derivatives, which are prevalent in calculus and advanced algebra courses.
For example, in our exercise, we start with a complex fraction and simplify it step by step. Initially, by expanding and subtracting terms within the numerator, and finally, by canceling out common factors in both the numerator and the denominator. In this case, the 'h' terms cancel each other out, leaving a constant as the simplified result of the difference quotient.
In practice, simplifying an algebraic expression is the right approach to prep the terrain for further mathematical procedures, such as taking limits or derivatives, which are prevalent in calculus and advanced algebra courses.
Limit Concepts
The concept of a limit is a foundational idea in calculus. It describes the behavior of a function as it approaches a certain point from either direction on the x-axis. Limits are essential for defining derivatives and integrals, making them a key concept for understanding the rate of change and the area under a curve, respectively.
In our exercise, while we have not directly computed a limit, the process of simplifying the difference quotient is a preparatory step for finding derivatives using the limit definition. Should we pursue the process further into calculus, we would apply the limit as 'h' approaches zero to the difference quotient. The value that the difference quotient approaches as 'h' becomes infinitesimally small is, in fact, the derivative of the function at a point 'x'.
Grasping the concept of limits allows students to better understand the behavior of functions, especially near points of interest, and gives rise to powerful applications in both mathematics and the sciences.
In our exercise, while we have not directly computed a limit, the process of simplifying the difference quotient is a preparatory step for finding derivatives using the limit definition. Should we pursue the process further into calculus, we would apply the limit as 'h' approaches zero to the difference quotient. The value that the difference quotient approaches as 'h' becomes infinitesimally small is, in fact, the derivative of the function at a point 'x'.
Grasping the concept of limits allows students to better understand the behavior of functions, especially near points of interest, and gives rise to powerful applications in both mathematics and the sciences.
Other exercises in this chapter
Problem 56
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