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Chapter 24: Problem 22

Solve the given problems involving tangent and normal lines. Where does the normal line to the parabola \(y=x-x^{2}\) at (1,0) intersect the parabola other than at (1,0)\(?\)

Short Answer

Expert verified
The normal line intersects the parabola at the point \((-1, -2)\) other than \((1,0)\).

Step by step solution

01

Find the Derivative of the Parabola

To determine the slope of the tangent line to the parabola, we start by differentiating the function. The parabola is given by the equation \(y = x - x^2\). The derivative \(y'\) of the function is found by differentiating each term: \(y' = \frac{d}{dx}(x - x^2) = 1 - 2x\).
02

Evaluate the Slope at the Given Point

Substitute \(x = 1\) into the derivative to find the slope of the tangent line at \((1,0)\). This gives us: \(y' = 1 - 2(1) = -1\). Thus, the slope of the tangent line at \((1,0)\) is \(-1\).
03

Determine the Slope of the Normal Line

The normal line to a curve at a given point is perpendicular to the tangent line at that point. Therefore, the slope of the normal line is the negative reciprocal of the slope of the tangent line. The slope of the normal line is thus \(1\), since the negative reciprocal of \(-1\) is \(1\).
04

Write the Equation of the Normal Line

Using point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is \((1,0)\) and \(m = 1\), the equation of the line becomes: \(y - 0 = 1(x - 1)\), simplifying to \(y = x - 1\).
05

Set the Equations Equal and Solve for Intersection

To find the intersection points, set the equation of the normal line equal to that of the parabola: \(x - 1 = x - x^2\). Simplify and solve:\[x - 1 = x - x^2\]This simplifies to:\[-1 = -x^2\]\[x^2 = 1\]Taking the square root of both sides, we find \(x = 1\) or \(x = -1\).
06

Verify the Intersection Points

When \(x = 1\), the point is \((1,0)\), which we already know. For \(x = -1\), substitute back into the original parabola equation to find the y-coordinate: \(y = -1 - (-1)^2 = -1 - 1 = -2\). Therefore, the other point of intersection is \((-1, -2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Lines
Tangent lines are straight lines that touch a curve at a single point. This point is called the point of tangency, and at this point, the slope of the tangent line is equal to the derivative of the curve. Specifically, for the equation of a curve given in the form \( y = f(x) \), the tangent line at a point \( (x_1, y_1) \) has a slope that is equal to \( f'(x_1) \).
  • This slope gives us the rate at which the curve is changing at \( x_1 \).
  • It essentially "hugs" the curve closely at this point, providing the best linear approximation.
For example, consider the parabola \( y = x - x^2 \). To find the tangent line at any point \( x \), you take the derivative, which calculates to \( 1 - 2x \). This represents the slope of the tangent line at any point \( x \) on the parabola.
Normal Lines
Normal lines are perpendicular to tangent lines at the point of tangency. If you know the slope of the tangent line, the slope of the normal line is the negative reciprocal. This is because the product of the slopes of two perpendicular lines is \(-1\).
  • To find the slope of the normal line, use the formula \( m_{normal} = -\frac{1}{m_{tangent}} \).
  • Normal lines are used in various applications like determining the direction of light reflection.
In our example, the parabola \( y = x - x^2 \) has a tangent line at point \( (1, 0) \) with a slope \(-1\). The normal line, therefore, has a slope of \( 1 \). This is calculated as \(-1\)'s negative reciprocal.
Derivative
A derivative measures how a function changes as its input changes. It's the "rate of change" of the function and is key to finding tangent lines. For a function \( f(x) \), the derivative \( f'(x) \) gives the function's slope at any point.
  • Basic rules: \( (x^n)' = nx^{n-1} \) and constant functions have a derivative of 0.
  • To find where functions increase or decrease, look at the sign of the derivative.
In our exercise, taking the derivative of the parabola \( y = x - x^2 \) gives us \( y' = 1 - 2x \). This tells us the slope of the tangent line at any \( x \). Evaluating at \( x = 1 \), \( y' = -1 \) tells us exactly how steep the tangent is at point \( (1, 0) \).
Equation of a Line
Creating the equation for a line relies on knowing a point on the line and the line's slope. This is most easily done using the point-slope form: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.
  • Transform this into slope-intercept form \( y = mx + b \) for easy graphing.
  • Essential in solving for intersections with other curves or lines.
In our case, we use the normal line's slope 1 from point \( (1, 0) \). Inserting these into point-slope form: \( y - 0 = 1(x - 1) \). This simplifies into \( y = x - 1 \), providing a clear linear equation for graphing or further analysis.

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Most popular questions from this chapter

Solve the given problems by finding the appropriate differential. If a spacecraft circles the earth at an altitude of \(250 \mathrm{km},\) how much farther does it travel in one orbit than an airplane that circles the earth at a low altitude? The radius of the earth is \(6370 \mathrm{km}\).

In Exercises 33 and \(34,\) view the graphs of \(y, y^{\prime}, y^{\prime \prime}\) together on a calculator. State how the graphs of \(y^{\prime}\) and \(y^{\prime \prime}\) are related to the graph of \(y\). $$y=x^{3}-12 x$$

Factories \(A\) and \(B\) are \(8.0 \mathrm{km}\) apart, with factory \(B\) emitting eight times the pollutants into the air as factory \(A .\) If the number \(n\) of particles of pollutants is inversely proportional to the square of the distance from a factory, at what point between \(A\) and \(B\) is the pollution the least?

Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator. The solar-energy power \(P\) (in W) produced by a certain solar system does not rise and fall uniformly during a cloudless day because of the system's location. An analysis of records shows that \(P=-0.45\left(2 t^{5}-45 t^{4}+350 t^{3}-1000 t^{2}\right),\) where \(t\) is the time (in h) during which power is produced. Show that, during the solar-power production, the production flattens (inflection) in the middle and then peaks before shutting down. (Hint: The solutions are integral.)

In Exercises 33 and \(34,\) view the graphs of \(y, y^{\prime}, y^{\prime \prime}\) together on a calculator. State how the graphs of \(y^{\prime}\) and \(y^{\prime \prime}\) are related to the graph of \(y\). $$y=24 x-9 x^{2}-2 x^{3}$$
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