Problem 4
Question
Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=5-2 x, \quad \text { at }(-3,11)$$
Step-by-Step Solution
Verified Answer
The slope of the tangent line is -2.
1Step 1: Understand the Problem
We need to find the slope of the tangent line to the function \( f(x) = 5 - 2x \) at the point \((-3, 11)\). The slope of the tangent line is the derivative of the function at the given point.
2Step 2: Find the Derivative of the Function
The function is \( f(x) = 5 - 2x \). To find the slope of the tangent line, we need to find its derivative. The derivative \( f'(x) \) represents the slope of the tangent to the graph of the function at any point \( x \). Using the power rule and the derivative of a constant, we get:\[ f'(x) = 0 - 2 = -2. \]
3Step 3: Evaluate the Derivative at the Given Point
The derivative \( f'(x) = -2 \) is constant, meaning the slope of the tangent line is \(-2\) at any point on the graph, including at \( x = -3 \). Therefore, the slope of the tangent line at \((-3, 11)\) is also \(-2\).
Key Concepts
Derivative of a FunctionPower RuleEvaluating Derivatives
Derivative of a Function
The derivative of a function is an essential concept in calculus. It provides a measure of how the function value changes as its input changes. In simpler terms, it tells us the slope of the tangent line at any given point on the graph of a function. This is crucial when we want to know the behavior of a function, like whether it is increasing or decreasing, and how steeply.
To find the derivative of a function, we use different rules that apply to various types of functions, one of which is the power rule. Once the derivative is calculated, it can be evaluated at any specific point to find the exact slope of the tangent line at that point.
To find the derivative of a function, we use different rules that apply to various types of functions, one of which is the power rule. Once the derivative is calculated, it can be evaluated at any specific point to find the exact slope of the tangent line at that point.
Power Rule
The power rule is a quick and straightforward way to find the derivative of a polynomial function. If you have a function of the form \(f(x) = ax^n\) where \(a\) is a constant and \(n\) is a positive integer, the derivative is given by \(f'(x) = nax^{n-1}\). This means you multiply the power by the coefficient and then decrease the power by one.
In our example, the function is \(f(x) = 5 - 2x\). This is already a simple one-term polynomial function \(-2x^1\). Using the power rule, the derivative is simply the coefficient of \(-2\), since the power of \(x\) is \(1\), which becomes \(0\) when decreased by one. So the derivative \(f'(x)\) is \(-2\). This rule helps in quickly determining the rate of change for any polynomial function.
In our example, the function is \(f(x) = 5 - 2x\). This is already a simple one-term polynomial function \(-2x^1\). Using the power rule, the derivative is simply the coefficient of \(-2\), since the power of \(x\) is \(1\), which becomes \(0\) when decreased by one. So the derivative \(f'(x)\) is \(-2\). This rule helps in quickly determining the rate of change for any polynomial function.
Evaluating Derivatives
After finding the derivative of a function, evaluating it at a specific point tells us the slope of the tangent line at that point. It's like zooming in on a particular part of the function to see how it behaves exactly there.
In our case, the derivative of the function \(f(x) = 5 - 2x\) was found to be a constant \(-2\). This means, regardless of the \(x\)-value we choose, the slope of the tangent line is always \(-2\). So, at the point \((-3, 11)\), the slope is \(-2\).
In our case, the derivative of the function \(f(x) = 5 - 2x\) was found to be a constant \(-2\). This means, regardless of the \(x\)-value we choose, the slope of the tangent line is always \(-2\). So, at the point \((-3, 11)\), the slope is \(-2\).
- This tells us that the graph is a line with constant slope.
- It also confirms there's no variation in slope along its length.
- Evaluating derivatives like this is a powerful tool to understand the structure of linear and other types of functions.
Other exercises in this chapter
Problem 3
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