Problem 14
Question
In Exercises 13 and \(14,\) find the slope of the curve at the indicated point. $$f(x)=|x-2|\( at \)x=1$$
Step-by-Step Solution
Verified Answer
The slope of the curve at the point \(x=1\) is \(-1\).
1Step 1: Rewrite The Function Without Absolute Value
The absolute value of \(x-2\) can be written as two separate functions: \(f(x) = x-2\) when \(x \geq 2\) and \(f(x) = 2-x\) when \(x < 2\). As we are looking for the slope at \(x=1\), we will use the second function: \(f(x) = 2-x\).
2Step 2: Find The Derivative Of The Function
The derivative of this function \(f'(x)\) gives the slope of the curve at any point \(x\). In this case, the derivative of \(f(x) = 2 - x\) is \(f'(x) = -1\).
3Step 3: Evaluate The Slope At The Indicated Point
To find the slope at the point \(x=1\), simply substitute \(x=1\) into \(f'(x)\). However in this case, the derivative is constant, so \(f'(1) = -1\).
Key Concepts
DerivativesSlope of a CurvePiecewise Functions
Derivatives
In calculus, derivatives represent the rate of change of a function as its input changes.
They are essential for understanding how functions behave and are the foundation for many concepts, such as calculating slopes and predicting future behavior of models in physics, economics, and more.
- When calculating the derivative, you follow rules such as the power rule, product rule, and chain rule.
These help you compute the derivative based on different functions.
- For example, the derivative of a linear function like \(f(x) = ax + b\) is simply \(a\), because the slope of a straight line is constant.
To find the derivative of a piecewise function, you need to consider each 'piece' or segment of the function separately.
This approach helps determine how the function changes across different domains.
Once found, the derivative allows you to analyze the characteristics of the function, such as solving for critical points or identifying where the function is increasing or decreasing.
They are essential for understanding how functions behave and are the foundation for many concepts, such as calculating slopes and predicting future behavior of models in physics, economics, and more.
- When calculating the derivative, you follow rules such as the power rule, product rule, and chain rule.
These help you compute the derivative based on different functions.
- For example, the derivative of a linear function like \(f(x) = ax + b\) is simply \(a\), because the slope of a straight line is constant.
To find the derivative of a piecewise function, you need to consider each 'piece' or segment of the function separately.
This approach helps determine how the function changes across different domains.
Once found, the derivative allows you to analyze the characteristics of the function, such as solving for critical points or identifying where the function is increasing or decreasing.
Slope of a Curve
The slope of a curve at a specific point represents how steep the curve is at that instant.
Unlike straight lines, curves can change their steepness at every point, making slope analysis more complex.
- To find the slope at a specific point, you compute the derivative of the function and substitute the desired input value.
- Essentially, the slope tells us if a function is increasing, decreasing, or flat at any point we're interested in.
For our example \(f(x) = 2-x\) when \(x < 2\), the derivative \(f'(x) = -1\) is used.
This indicates that at every point within this interval, including \(x = 1\), the curve has a slope of \(-1\).
This means the function is decreasing consistently as \(x\) increases within this range.
Unlike straight lines, curves can change their steepness at every point, making slope analysis more complex.
- To find the slope at a specific point, you compute the derivative of the function and substitute the desired input value.
- Essentially, the slope tells us if a function is increasing, decreasing, or flat at any point we're interested in.
For our example \(f(x) = 2-x\) when \(x < 2\), the derivative \(f'(x) = -1\) is used.
This indicates that at every point within this interval, including \(x = 1\), the curve has a slope of \(-1\).
This means the function is decreasing consistently as \(x\) increases within this range.
Piecewise Functions
Piecewise functions are composed of multiple sub-functions, each defined on its own domain.
They help model situations where a function changes behavior at different intervals or points.
- These are useful in representing real-world scenarios, such as tax brackets or speed limits, where conditions depend on certain ranges.
- Each segment has its own rule and might show different characteristics.
To analyze a piecewise function, first rewrite the function to separate its components according to their applicable intervals.
Once segmented, you can differentiate each section to analyze behavior such as slopes or critical points.
For \(f(x) = |x-2|\), the function changes definition around \(x = 2\) and each segment is handled according to its rule.
They help model situations where a function changes behavior at different intervals or points.
- These are useful in representing real-world scenarios, such as tax brackets or speed limits, where conditions depend on certain ranges.
- Each segment has its own rule and might show different characteristics.
To analyze a piecewise function, first rewrite the function to separate its components according to their applicable intervals.
Once segmented, you can differentiate each section to analyze behavior such as slopes or critical points.
For \(f(x) = |x-2|\), the function changes definition around \(x = 2\) and each segment is handled according to its rule.
Other exercises in this chapter
Problem 13
In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x
View solution Problem 13
In Exercises 13-20, use graphs and tables to find the limits. $$\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}$$
View solution Problem 14
In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow 2 } \sqrt { x + 3 }$$
View solution Problem 14
In Exercises 13-20, use graphs and tables to find the limits. $$\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}$$
View solution