Problem 85 - A Graphical Approach to College Algebra

Question

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) ) for each function \(f\). Simplify completely. $$f(x)=1-x^{2}$$

Step-by-Step Solution

Answer

The simplified difference quotient is \(-2x - h\).

Step 1: Identify the given function and expressions

The function provided is \(f(x) = 1 - x^2\). The difference quotient we need to compute is given by \(\frac{f(x+h) - f(x)}{h}\).

Step 2: Compute \(f(x+h)\)

To find \(f(x+h)\), substitute \(x + h\) into the function \(f(x)\): \[f(x+h) = 1 - (x+h)^2\] Simplify this: \[f(x+h) = 1 - (x^2 + 2xh + h^2) \] \[f(x+h) = 1 - x^2 - 2xh - h^2\]

Step 3: Calculate \(f(x+h) - f(x)\)

Using \(f(x+h)\) and \(f(x)\), compute the expression: \[f(x+h) - f(x) = (1 - x^2 - 2xh - h^2) - (1 - x^2)\]Simplify it: \[f(x+h) - f(x) = 1 - x^2 - 2xh - h^2 - 1 + x^2\] \[f(x+h) - f(x) = -2xh - h^2\]

Step 4: Form the difference quotient

Substitute \(f(x+h) - f(x)\) back into the difference quotient: \[\frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2}{h}\].

Step 5: Simplify the difference quotient

Factor out \(h\) from the numerator: \[\frac{-2xh - h^2}{h} = \frac{h(-2x - h)}{h}\] Cancel \(h\) in the numerator and the denominator (since \(h eq 0\)), we get: \[-2x - h\] The simplified form of the difference quotient is \(-2x - h\).

Key Concepts

Quadratic Functions

Quadratic functions are a crucial concept in algebra and calculus. They are polynomial functions of degree two and are represented by the general form:

• \(f(x) = ax^2 + bx + c\),

where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.Quadratic functions often form a parabola when graphed. The orientation of the parabola (opening upwards or downwards) depends on the coefficient \(a\); if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.In the problem, the specific quadratic function is \(f(x) = 1 - x^2\). The coefficient of \(x^2\) is \(-1\), which means the parabola opens downwards. The vertex of this parabola is the point

• \((0, 1)\).

Understanding the behavior and properties of quadratic functions is vital because many real-world phenomena can be modeled using them. It also sets a foundation for learning complex calculus concepts like differentiation and integration.

Algebraic Simplification

Algebraic simplification involves rewriting an expression in a simpler or more convenient form without changing its value. This process often includes

• expanding expressions,
• factorization,
• combining like terms, and
• canceling terms.

In the exercise, we see this simplification first hand during Steps 2 and 3.To find \(f(x+h)\), we substitute \(x+h\) into the function to get \(1 - (x + h)^2\). We then expand the expression:

• \((x + h)^2 = x^2 + 2xh + h^2\).

Substituting and simplifying leads to removing unnecessary terms and combining like terms until you are left with

• \(-2xh - h^2\).

Finally, when forming the difference quotient: \[ \frac{-2xh - h^2}{h} \]we factor out \(h\) from the numerator and cancel with the denominator, assuming \(h eq 0\), to arrive at the simple result of

• \(-2x - h\).

This understanding is essential because it enables clearer insights when working with more complicated algebraic equations.

Limits in Calculus

Limits form a fundamental aspect of calculus and are essential for understanding derivatives and integrals. The concept of a limit involves finding the value that a function approaches as the input (or variable) approaches a particular value.In the context of the difference quotient, limits help in identifying the instantaneous rate of change of a function. When calculating the derivative of a function, the difference quotient

• \( \frac{f(x + h) - f(x)}{h} \)

is taken in the limit as \(h\) approaches zero.For a quadratic function like in the problem, applying the limit to our simplified difference quotient,

• \(-2x - h\)

as \(h \rightarrow 0\), results in the derivative

• \(-2x\).

This reveals the slope of the tangent line at any point \(x\) on the curve of the quadratic function \(f(x) = 1 - x^2\). Understanding how limits work with the difference quotient is critical for mastering calculus, as it unlocks the ability to explore changes and dynamics in various functions.