Determine the slope of the tangent to each of the following curves:
1) y=2/x^2
2) y=8x-2x^2
3) y=x^4-2/x.

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See the attached file.

The slope of a line is a measure of how "steep" it is. When you climb a steep hill, you're climbing a hill with great slope. Level ground has a zero slope, and something entirely vertical, like a wall, is said to have infinite slope. When we climb a hill, we are, in a sense, going forward as well as up. Slope measures how much we go up as we go forward. Zero slope means we don't go up at all, not matter how much we go forward. This is what happens on level ground. If we're scaling a wall, we're going up without going forward at all. We're going up from the same spot on the ground. Thus we may say that a wall has infinite slope. With any case in between, we're both going up and forward at the same time.

But these are only intuitive ideas. In order to speak of them more rigorously, we need to define slope mathematically.
See attached image file - "Slope.png"
Given a straight line, consider a section of it, as shown in the figure. Since we want to measure ...

Solution Summary

This solution explains the concept and mathematical theory of the slope of a curve and shows how it can be calculated for any curve defined by a Cartesian equation, using differentiation. An explanation of the theory is followed by three examples. Links for further reference are also included.

... I) Find the slope of the tangent line to the given polar curve at the point specified by the value theta. ...Tangents to polar curves are investigated. ...

... since dy/dx means the slope of a tangent to the curve at a certain point. ... since dy/dx means the slope of a tangent to the curve at a certain point. ...

... First, we find the slopes of the tangents, and use this along with the points given to find the equations ... Then we use the slope of the tangent to calculate ...

... Using the expressions (1) and (2), we can write (3) Thus, we proved that the slope of the tangent at the curve in the point (A) defined by the abscissa (x0) is ...

... We got the same slope as in equation (1.5), and of course ... in equation (1.7), thus the equation of the tangent line is ... It looks like this: The parametric curve is ...

Use implicit differentiation to find the slope of the curve : x^3 - 3xy + y^4 = 5x at (0.5, -0.977). Find the equation of the tangent line at this point. ...

... So, the slope of the tangent line for the curve is given by . It is given that the tangent line has given slope =9. Hence we should have This gives . ...

... 0 +3 = 3 F'(1) = 6*(1)^2 - 8*1 + 3 = 1 They are the slopes of the tangent lines to ... point (0,-5). The normal line to the curve at the point has the slope -1/f ...

... e. Use the derivative to calculate the slope of the tangent line (the ... Derivatives, Tangents and Interpreting the Slope of a Regression Line are investigated ...