Problem 2 - Calculus with Concepts in Calculus

Question

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x-3 x^{2} ; a=-1, b=3 $$

Step-by-Step Solution

Answer

The number \(c = 1\) is the point in the interval \((-1, 3)\) where the tangent line is parallel to the chord.

Step 1: Understand the Problem

The task is to find points \(c\) in the interval \((a, b)\) where the line tangent to the function \(f(x) = x - 3x^2\) is parallel to the line joining the points \((a, f(a))\) and \((b, f(b))\). This means we need to find \(c\) such that the slope of the tangent at \(c\) equals the slope of this chord.

Step 2: Calculate the Slope of the Chord

Determine \(f(a)\) and \(f(b)\) first: \(f(-1) = -1 - 3(-1)^2 = -4\)\(f(3) = 3 - 3(3)^2 = -24\).The slope \(m\) of the line joining \((-1, -4)\) and \((3, -24)\) is given by:\[ m = \frac{f(3) - f(-1)}{3 - (-1)} = \frac{-24 - (-4)}{3 - (-1)} = \frac{-20}{4} = -5. \]

Step 3: Calculate the Derivative of f(x)

To find the slope of the tangent line, calculate the derivative \(f'(x)\):\[ f'(x) = \frac{d}{dx}(x - 3x^2) = 1 - 6x. \]

Step 4: Set Derivative Equal to Chord Slope

Set the derivative equal to the slope from Step 2 to find \(c\):\[ 1 - 6c = -5. \]

Step 5: Solve for c

Solve the equation from Step 4 for \(c\):\[ 1 - 6c = -5 \]\[ -6c = -5 - 1 \]\[ -6c = -6 \]\[ c = 1. \]

Step 6: Verify c is Within the Interval

Check that \(c = 1\) is within the interval \((-1, 3)\). Since 1 is indeed within the range, \(c = 1\) is a valid solution.

Key Concepts

Tangent Lines

In calculus, a tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. The line is "tangent" in the sense that it has the same slope as the curve does at the point of contact.
This means the tangent line has the same direction as the curve at that specific point; they are "parallel" to one another.
When dealing with functions, you'll often find the equation of a tangent line by finding the derivative of the function, which tells you how steep the graph is at any particular point. The tangent at any point can be expressed as a simple equation of a line:

• The slope of the tangent.
• The point where it touches the curve.

Understanding tangent lines is crucial because they provide a linear approximation of a function at a specific point, making complex functions more manageable for analysis and predictions.

Derivatives

Derivatives are a fundamental concept in calculus, acting as a tool to measure how fast a function is changing at any given point. Essentially, the derivative tells you the slope of the tangent line to the curve defined by a function at any point.

• The process of finding a derivative is called differentiation.
• For example, if you have a function like \[f(x) = x - 3x^2,\] finding its derivative gives you \[f'(x) = 1 - 6x.\]

With this derivative, \(f'(x)\), you can find the slope of the tangent line at any point on the curve. This slope is also referred to as the instantaneous rate of change.
Derivatives are essential for understanding many aspects of functions, such as identifying critical points where function behavior changes (like maximum and minimum values). They are a staple of calculus because they help us analyze how functions behave and give insights into the geometric properties of curves.

Slopes of Curves

Slopes of curves are a key idea in understanding any graphical representation of functions. When we refer to the slope of a curve, we are generally concerned with the steepness or inclination of the curve at a particular point.
For instance, when asked to find a point where the slope of a tangent line is equal to the slope of a chord (a straight line connecting two points on the curve), we are essentially searching for parallelism between the lines.

• Calculating the slope of a chord involves simple subtraction of the \(y\)-values divided by subtraction of the \(x\)-values, as in the formula\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]
• For a dynamic understanding of curve slopes, derivatives are utilized, signifying changes along infinitesimally small segments.

Understanding the slope of curves is vital to solving many calculus problems and visualizing how functions behave as they plot on a graph.

Intervals in Calculus

Intervals in calculus refer to the continuous range of values that a variable, like \(x\), can take. These are typically expressed in terms of open or closed intervals.

• An open interval, expressed as \((a, b)\), indicates that the endpoints\(a\) and \(b\) are not included.
• A closed interval, shown as \([a, b]\), means both endpoints are included.

When solving calculus problems, identifying the right interval is crucial as it defines the domain where you're solving for solutions or making analyses.
For example, to find \(c\) in an interval \((a, b)\) where a tangent line is parallel to a chord, the value of \(c\) has to be checked to see if it is included in that interval.
Knowing intervals ensures calculations are appropriately constrained and solutions are verified within the given parameters. This is crucial when interpreting real-world problems through calculus models, ensuring solutions remain valid.