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Chapter 1: Problem 13

Find an equation of the given line. \((0,0)\) and \((1,0)\) on line

Short Answer

Expert verified
The equation of the line is \ y = 0 \.

Step by step solution

01

- Identify Points

The given points are \( (0,0) \) and \( (1,0) \).
02

- Determine the Slope

The slope (m) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{0 - 0}{1 - 0} = 0 \]
03

- Use the Point-Slope Form

The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting \((x_1, y_1) = (0,0)\) and \ m = 0 \, we get: \[ y - 0 = 0(x - 0) \] Simplifying, we obtain: \[ y = 0 \]

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Key Concepts

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

Determining the slope
The slope of a line provides a measure of its steepness and direction. To find the slope, you need two points on the line. Once you have your points, use the formula for the slope: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.In the given problem, these points are \((0, 0)\) and \((1, 0)\). Using the slope formula:\[m = \frac{0 - 0}{1 - 0} = 0\]This tells us the line is horizontal, and a horizontal line always has a slope of zero.
  • Slope lets you know how fast and in which direction a line goes up or down.
  • A zero slope means no rise, just a run (horizontal line).
Knowing how to calculate the slope is essential to understanding how a line behaves.
Point-Slope Form
Once you know the slope, you can use the point-slope form of the equation of a line, which comes in handy when you know a single point and the slope.The equation is:\[y - y_1 = m(x - x_1)\]Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.From our example, we have the point \((0, 0)\) and a slope of 0, so we substitute these values into the formula:\[y - 0 = 0(x - 0)\]This simplifies to:\[y = 0\]
  • Point-slope form makes it easier to write the equation of the line quickly.
  • This form is especially useful when you have tricky points or a non-zero slope.
Remember, the point-slope form is a powerful tool for transitioning from a known slope and point to the full equation of the line.
Simplifying Equations
Finally, simplifying the equation is crucial. The goal is to reduce the equation into the simplest form. For the given problem, we started with the point-slope form:\[y - 0 = 0(x - 0)\]Since anything multiplied by zero is zero, this simplifies immediately to:\[y = 0\]This is the simplest form of the equation and directly tells us that for all x-values, y is always zero. Always remember:
  • Simplified equations are clearer and easier to work with.
  • Showing each step of simplification helps avoid errors and makes your process transparent.
When solving, always aim to simplify. This not only makes equations easier to read but also reveals important properties of the line.

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Most popular questions from this chapter

(a) Draw two graphs of your choice that represent a function \(y=f(x)\) and its vertical shift \(y=f(x)+3\). (b) Pick a value of \(x\) and consider the points \((x, f(x))\) and \((x, f(x)+3)\). Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines. (c) Based on your observation in part (b), explain why $$ \frac{d}{d x} f(x)=\frac{d}{d x}(f(x)+3) . $$

After an advertising campaign, the sales of a product often increase and then decrease. Suppose that \(t\) days after the end of the advertising, the daily sales are \(f(t)=-3 t^{2}+32 t+100\) units. What is the average rate of growth in sales during the fourth day, that is, from time \(t=3\) to \(t=4 ?\) At what (instantaneous) rate are the sales changing when \(t=2 ?\)

Consider the cost function \(C(x)=6 x^{2}+14 x+18\) (thousand dollars). (a) What is the marginal cost at production level \(x=5 ?\) (b) Dstimate the cost of raising the production level from \(x=5\) to \(x=5.25\) (c) Let \(R(x)=-x^{2}+37 x+38\) denote the revenue in thousands of dollars generated from the production of \(x\) units. What is the break-even point? (Recall that the break-even point is when revenue is equal to cost.) (d) Compute and compare the marginal revenue and marginal cost at the break- even point. Should the company increase production beyond the break-even point? Justify your answer using marginals.

Rates of Change Suppose that \(f(x)=-6 / x .\) (a) What is the average rate of change of \(f(x)\) over each of the intervals 1 to 2,1 to \(1.5\), and 1 to \(1.2 ?\) (b) What is the (instantaneous) rate of change of \(f(x)\) when \(x=1 ?\)

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=x^{1 / 5}\)
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