The Tangent Line Problem and the Min/Max Problem in Calculus

FINDING THE EQUATION OF THE TANGENT LINE TO A CURVE

In order to find the equation of the tangent line to the curve at a given point you  need two things: 1) the slope of the line at that point, and 2) the (x1, y1) coordinates of the particular point.  Typically you are given the function and the x1 value where you want the tangent line to touch the curve.

By taking the derivative of the original function, you get another function of x.  By subsituting the given value of x, which we are calling x1, into the derivative function, you can find the value of the slope of the tangent line at that point.  We call the value of the slope m.

You then have to find the y value that corresponds to the given x1, which we are calling y1.  To do that, you substitute x1 into the original given function to get the y1 value that corresponds to that given x1.

You now have the slope of the tangent line, and a point the line goes through.  By substituting into the following equation of the straight line you get the equation of the tangent line at point (x1, y1).

(y-y1) = m (x-x1)

THE MAXIMUM-MINIMUM PROBLEM

The key to the minimum/maximum problem in calculus is to realize that at the peak or valley of a curve, the tangent line is horizontal, i.e., the slope of the tangent line is zero.

To find the x value that corresponds to the max. or min., take the original function, and find its derivative, and set it equal to zero.  Solve for x.  That gives you the x value corresponding to the max or min.

Then if you substitute that x back into the original function, you get the value of the maximum value or minimum value of the function.  In the case of the parabola, we know that if the leading coefficient is positive, then it is upward facing, and if the leading coefficient is negative, it is downward facing.

For example, an upward facing (think +x squared) parabola has a min, and a downward facing parabola (think -x squared) has a max.  [Later on in Calculus we will learn another method to figure this out analytically using second derivatives.]

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