For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1-x^2

Verified step by step guidance

1. Step 1: Start by finding \( f(x+h) \). Substitute \( x+h \) into the function \( f(x) = 1 - x^2 \) to get \( f(x+h) = 1 - (x+h)^2 \).
2. Step 2: Expand \( (x+h)^2 \) to get \( x^2 + 2xh + h^2 \). Therefore, \( f(x+h) = 1 - (x^2 + 2xh + h^2) \).
3. Step 3: Simplify \( f(x+h) \) to \( 1 - x^2 - 2xh - h^2 \).
4. Step 4: Find \( f(x+h) - f(x) \) by subtracting \( f(x) = 1 - x^2 \) from \( f(x+h) = 1 - x^2 - 2xh - h^2 \). This results in \( -2xh - h^2 \).
5. Step 5: Divide \( f(x+h) - f(x) \) by \( h \) to find \( \frac{f(x+h) - f(x)}{h} \). Simplify \( \frac{-2xh - h^2}{h} \) to get \( -2x - h \).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, evaluating ƒ(x+h) means replacing 'x' in the function ƒ(x) = 1 - x² with 'x+h'. This process is fundamental in calculus and algebra as it lays the groundwork for understanding how functions behave as their inputs change.

Difference Quotient

The difference quotient is a formula that represents the average rate of change of a function over an interval. It is calculated as [ƒ(x+h) - ƒ(x)]/h, where 'h' is the change in 'x'. This concept is crucial for understanding derivatives in calculus, as it approximates the slope of the tangent line to the function at a point.

Limit Concept

The limit concept is essential in calculus, describing the behavior of a function as its input approaches a certain value. In the context of the difference quotient, as 'h' approaches zero, the expression [ƒ(x+h) - ƒ(x)]/h approaches the derivative of the function at 'x'. This concept is foundational for defining derivatives and understanding instantaneous rates of change.