Problem 14
Question
Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=-\frac{1}{2} x^{2}\)
Step-by-Step Solution
Verified Answer
In conclusion, using the four-step process, the slope of the tangent line to the graph of the function \(f(x) = -\frac{1}{2}x^{2}\) at any point x is found by taking the derivative, \(f'(x) = -x\), and the slope at any point x is \(-x\).
1Step 1: Determining the Point on the Graph
We want to find the slope of the tangent line at any given point for the function \(f(x) = -\frac{1}{2}x^{2}\). So, we will only need to focus on the x-value and use it later when finding the slope.
2Step 2: Finding the Derivative of the Function
To find the derivative of the function, we will use the power rule, which states that if \(f(x) = ax^n\), then \(f'(x) = nax^{n-1}\).
Applying the power rule to our function \(f(x) = -\frac{1}{2}x^{2}\), we get:
\(f'(x) = 2(-\frac{1}{2})x^{2-1} = -x\)
3Step 3: Plugging in the x-value of the Point into the Derivative
As we are trying to find the slope of the tangent line at any point, we can use \(x\) as a general representation of the x-value to plug into the derivative. So, we have \(f'(x) = -x\)
4Step 4: Finding the Slope of the Tangent Line
The derivative of the function at any given point will give us the slope of the tangent line. In our case, we found that the derivative is \(f'(x) = -x\). This means that the slope of the tangent line at any point x on the graph will be equal to \(-x\).
In conclusion, the slope of the tangent line to the graph of the function \(f(x) = -\frac{1}{2}x^{2}\) at any point x is \(-x\).
Key Concepts
DerivativesTangent LinePower Rule
Derivatives
Derivatives are a fundamental aspect of calculus. They represent the rate at which a function is changing at any given point. This concept is especially important when analyzing graphs because it reveals the slope of the graph at a specific point.
To find a derivative, one must understand the original function and then apply certain rules, often beginning by considering the function in terms of its variables. In general, if the function is represented as \( f(x) \), the derivative is noted as \( f'(x) \). This new function, the derivative, gives the slope or rate of change of \( f(x) \) at each point.
Derivatives can be geometrically interpreted as the slope of the tangent line to the curve at any point. Simply put, they show how much a function increases or decreases as the input changes slightly. Calculating derivatives involves some common rules, including the sum rule, product rule, chain rule, and most frequently in basic calculations, the power rule.
To find a derivative, one must understand the original function and then apply certain rules, often beginning by considering the function in terms of its variables. In general, if the function is represented as \( f(x) \), the derivative is noted as \( f'(x) \). This new function, the derivative, gives the slope or rate of change of \( f(x) \) at each point.
Derivatives can be geometrically interpreted as the slope of the tangent line to the curve at any point. Simply put, they show how much a function increases or decreases as the input changes slightly. Calculating derivatives involves some common rules, including the sum rule, product rule, chain rule, and most frequently in basic calculations, the power rule.
Tangent Line
The tangent line is a straight line that touches a curve at a single point without intersecting it. It best represents the behavior of the curve at that specific point.
In calculus, when we want to talk about changing rates or slopes at a particular point, we're often talking about the tangent line at that point. This helps in understanding the immediate rate of change of the curve.
A tangent line is not just a touch-and-go concept; its slope is tangent to the slope of the curve itself at that particular point. The derivative of the function at a certain point provides the slope for this tangent line. This is why finding the derivative is key when determining the precise characteristics of the tangent line.
Thus, understanding tangent lines is crucial for solving various problems related to instantaneous rates of change and for approximating the function linearly at a local point, which brings smoothness and precision to calculations involving curves.
In calculus, when we want to talk about changing rates or slopes at a particular point, we're often talking about the tangent line at that point. This helps in understanding the immediate rate of change of the curve.
A tangent line is not just a touch-and-go concept; its slope is tangent to the slope of the curve itself at that particular point. The derivative of the function at a certain point provides the slope for this tangent line. This is why finding the derivative is key when determining the precise characteristics of the tangent line.
Thus, understanding tangent lines is crucial for solving various problems related to instantaneous rates of change and for approximating the function linearly at a local point, which brings smoothness and precision to calculations involving curves.
Power Rule
The power rule is a simple yet powerful tool in calculus used to find the derivative of polynomial functions. Knowing when and how to use it effectively allows for straightforward derivative computations.
Specifically, the power rule is applied when a function is expressed as \( f(x) = ax^n \). To differentiate such a function, you multiply the exponent \( n \) by the coefficient \( a \), and subtract one from the exponent, resulting in the formula \( f'(x) = nax^{n-1} \).
For example, with the function \( f(x) = -\frac{1}{2}x^2 \), the power rule lets us easily find that \( f'(x) = -2 \times \frac{1}{2}x^{2-1} = -x \). This shows the new function \( f'(x) \) which defines the slope of the original function at each point.
Specifically, the power rule is applied when a function is expressed as \( f(x) = ax^n \). To differentiate such a function, you multiply the exponent \( n \) by the coefficient \( a \), and subtract one from the exponent, resulting in the formula \( f'(x) = nax^{n-1} \).
For example, with the function \( f(x) = -\frac{1}{2}x^2 \), the power rule lets us easily find that \( f'(x) = -2 \times \frac{1}{2}x^{2-1} = -x \). This shows the new function \( f'(x) \) which defines the slope of the original function at each point.
- Helps in finding slopes of curves at any point.
- Reduces complex polynomial equations into manageable derivatives.
- Provides a foundation for more advanced differentiation techniques.
Other exercises in this chapter
Problem 14
Find the derivative of each function. \(f(x)=\left(x^{3}+2 x+1\right)\left(2+\frac{1}{x^{2}}\right)\)
View solution Problem 14
Find the derivative of the function \(f\) by using the rules of differentiation. \(f(u)=\frac{2}{\sqrt{u}}\)
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Complete the table by computing \(f(x)\) at the given values of \(x\). Use these results to estimate the indicated limit (if it exists). $$ \begin{array}{l} f(x
View solution Problem 15
Find the derivative of each function. \(f(x)=\frac{1}{(2 x+3)^{3}}\)
View solution