Find Slope of Line & Solve Approximation Mistake: Answers Here

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In summary, the slope of the line containing the point (4,0) and tangent to the graph of y=e^(-x) is -0.050. For the first question, the mistake was substituting x=4 instead of x=5. For the second question, it was using the left-end point method instead of the right-end point method.
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Find the slope of the line containing the point (4,0) and tangent to the graph of y=e^(-x) : I found the derivative (-e^(-x)) and substituted x=4. I got an answer of -0.018 but the answer is -0.050.. what did i do wrong?If http://texify.com/img/%5CLARGE%5C%21%5Cint_%7B0%7D%5E%7B3%7D%20%28x%5E2%20-4x%20%2B4%29dx%20.gif is approximated by 3 inscribed rectangles of equal width on the x-axis, then the approximation is : I got the answer of 5 but it is apparently 1. I did : 1(4)+ 1(1)+(1)(0) where the width of the rectnagles is 1 and and the second values are the respective y values at x.

What mistake did I make here as well?
 
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The tangent line passes through (4,0). It is not tangent to the curve when x= 4. Assume the line is tangent to the curve at, say, x= a and write the equation for the tangent line in terms of a. Set x= 4 and y= 0 and solve for a.

As far as your second question is concerned, I don't see how we can say what you did wrong when you don't say what you did! Of course, the result depends upon exactly where you evaluate the function in each interval so there are an infinite number of correct answers. What do you get if you choose the right endpoint of each interval to evaluate the function?
 
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razored said:
I got the answer of 5 but it is apparently 1. I did : 1(4)+ 1(1)+(1)(0) where the width of the rectnagles is 1 and and the second values are the respective y values at x.
I was told to use 3 rectangles of equal width on the x-axis implying delta x =1 for each rectangle. Then I used the left-end point method. Consequently, (1)f(0) + 1f(1) + 1f(2) = 1(4)+ 1(1)+(1)(0) = 5 but the answer is 1. I don't see why.


For the first one, i did http://texify.com/img/%5CLARGE%5C%21%5Cfrac%7Be%5E%7B-x%7D%7D%7Bx-4%7D%3D-e%5E%7B-x%7D.gif which[/URL] I found x to be 5. I substituted 5 into -e^(-x) and that does not equal -.05 . What did I do wrong now?
 
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FAQ: Find Slope of Line & Solve Approximation Mistake: Answers Here

1. What is slope and why is it important?

Slope is a measure of the steepness of a line. It is important in mathematics and science because it helps us understand and describe the relationship between two variables. In physics, slope is used to calculate velocity and acceleration, and in chemistry, it is used to determine reaction rates.

2. How do you find the slope of a line?

The slope of a line can be found by calculating the change in the y-values (vertical change) divided by the change in the x-values (horizontal change) between two points on the line. This is represented by the formula: slope = (y2 - y1) / (x2 - x1).

3. What is an approximation mistake?

An approximation mistake, also known as a rounding error, is a mistake made when a number is rounded to a certain decimal place or significant figure. This can result in a small difference between the actual value and the approximate value. In mathematics and science, it is important to be aware of and account for approximation mistakes in order to ensure accurate calculations and results.

4. How do you solve an approximation mistake?

To solve an approximation mistake, you can either use a more precise method of calculation or increase the number of decimal places or significant figures used in the approximation. It is important to be aware of the level of precision needed for a specific calculation and to use the appropriate number of decimal places or significant figures to avoid making an approximation mistake.

5. Can you give an example of finding slope of a line and solving an approximation mistake?

Sure, let's say we have the two points (2, 5) and (4, 11) on a line. To find the slope, we would use the formula: slope = (11 - 5) / (4 - 2) = 3. However, if we had rounded the y-values to the nearest whole number, we would have gotten a slope of 3.5 instead. To solve this approximation mistake, we can either use a more precise method of calculation (such as using more decimal places) or we can increase the level of precision in our approximation by using more decimal places for the y-values, resulting in a more accurate slope of 3.25.

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