Problem 23

Question

Find the slope of the tangent line to the graph of the given function at the given point $P$. $$ f(x)=3 x^{2}+6 \quad P=(-1,9) $$

Step-by-Step Solution

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Answer

The slope of the tangent line at point $(-1, 9)$ is $-6$.

1Step 1: Find the derivative of the function

To find the slope of the tangent line, we first need to determine the derivative of the function. Given the function $ f(x) = 3x^2 + 6 $, differentiate with respect to $ x $. The derivative, $ f'(x) $, represents the slope of the tangent line at any point $ x $ on the curve.Using the power rule ($ d/dx [x^n] = nx^{n-1} $), the derivative is calculated as follows:\[ f'(x) = d/dx [3x^2] + d/dx[6] = 6x + 0 = 6x \]

2Step 2: Evaluate the derivative at the point P

Now, substitute the $ x $-value of the point $ P $ into the derivative to find the slope of the tangent line at this specific point. The point $ P $ is given as $(-1, 9)$.Substitute $ x = -1 $ into $ f'(x) $:\[ f'(-1) = 6(-1) = -6 \]

3Step 3: Conclusion on the slope

The slope of the tangent line to the graph of the function at the point $ P = (-1, 9) $ is determined by the evaluated derivative from the previous step. Therefore, the slope is $ -6 $.

Key Concepts

DerivativeTangent LineSlopePower Rule

Derivative

A derivative is a fundamental concept in calculus. It's essentially a tool that helps us understand the rate at which a function is changing at any given point. Think of it as the "speed" of the function. For the function $ f(x) = 3x^2 + 6 $, the derivative is found by applying the rules of differentiation, specifically the power rule. The derivative $ f'(x) = 6x $ gives us the slope of the tangent line at any point $ x $ along the function. This means that as $ x $ changes, so does the slope of our tangent line.

The derivative tells us how steep our function is at any value of $ x $.
We use derivatives to find the instantaneous rate of change.
Differentiation is the process of finding a derivative.

Calculating derivatives is a crucial skill in calculus that allows us to analyze and interpret functions in detail.

Tangent Line

The tangent line is a straight line that "just touches" a curve at a specific point without crossing it. It's an approximation of the curve at that point, giving us a linear representation of how the curve behaves. In this problem, we're interested in the tangent line to the function $ f(x) = 3x^2 + 6 $ at the point $ P = (-1, 9) $.

A tangent line shares both a point and a slope with the function at that point.
The slope of the tangent line is obtained from the derivative of the function.

The equation of the tangent line can be expressed using the point-slope form of a line equation, $ y - y_1 = m(x - x_1) $, where $ m $ is the slope from the derivative, and $ (x_1, y_1) $ is the point $ P $. Thus, it acts as a linear snapshot of the nonlinear curve.

Slope

The slope describes the steepness and direction of a line. In the context of calculus, when we talk about slope, we are often referring to the slope of the tangent line to a curve at a given point. This is essentially the value of the derivative at that point.

For a line, slope is calculated as "rise over run", or the change in $ y $ over the change in $ x $.
In this case, the calculated slope at point $ P $ is $-6$, showing that the tangent line is relatively steep and decreases as $ x $ increases.

This means that at the point $ P = (-1, 9) $, the function decreases in $ y $-value by 6 units for every 1 unit increase in $ x $. Understanding slope helps visualize how the function behaves locally around $ P $.

Power Rule

The power rule is a straightforward, yet powerful tool used in calculus for differentiation. It makes finding derivatives of polynomial functions a breeze. The rule states that for a function $ f(x) = x^n $, the derivative is $ f'(x) = nx^{n-1} $.

The power rule simplifies the process of finding derivatives of terms like $ x^2 $ or $ x^3 $.
In this exercise, the power rule was used to find the derivative $ f'(x) = 6x $ from $ f(x) = 3x^2 + 6 $.
It allows us to quickly determine the slope of a function at any given $ x $ value.

This rule is a cornerstone of differentiation techniques and is essential for efficiently tackling calculus problems involving polynomial functions.

Problem 23

Other exercises in this chapter

Problem 23

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Problem 23

Use the Reciprocal Rule to compute the derivative of the given expression with respect to $x$ $$ 1 /(x+3 \cos (x)) $$

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Problem 23

A function $f$ and a value $c$ are given. Find an equation of the tangent line to the graph of $f$ at $(c, f(c))$. $$ f(x)=\sin (x), c=\pi / 3 $$

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Problem 24

Differentiate the given expression with respect to $x$. $$ \cos (\arctan (x)) $$

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