Prepared : Fri Sep 17 17:30:59 CDT 1999

MATH 1271

QUIZ I

1) Suppose that f is the function given by the formula

Using the slope-predictor formula for quadratic equations find the slope of the line tangent to y=f(x) at the point x=a. Then write an equation of the tangent line at the point (2, f(2)).

Remember that slope-predictor formula for the quadratic equation

is given as

Here one other small remainder will be that m(a) is the slope of the tangent line to f at the point (a, f(a)).(The point (a,f(a)) is on the graph of y=f(x)).

To avoid any possible disasters, let's take time to rewrite the definition of f(x).

In the rewritten form it is easier to find what p and q is:

Hence using the above cited slope-predictor formula we obtain

which is

in simpler terms. And this expression is also the slope of the tangent line to y=f(x) at the point x=a. So we finished the first part of the question.

To write down an equation of a line we either need to points or a point and the slope of the line. In the second case if is the point on the line and m is the slope, than the equation of the line is given by the following formula:

A tangent line to y=f(x) at the point x=a has the property that is passes through the point (a,f(a)) on the graph, and has slope=m(a). Let's find what all these terms become in our case:

Using the above slope-point equation, we get

2) Find the maximum possible value of the product of two positive numbers whose sum is 50.

Since it is common to use variables in mathematics, let's start with picking up to letters which will be our numbers, say x,y are the two numbers whose sum is 50 (in symbols: x+y=50). What we want to show is to find the maximum value of xy for all possible x,y. If you had a chance to try maximizing problems before, you will guess that it is not easy to maximize a function in two variables. So we should get rid of one of those variables, by being clever. Remember the condition on x and y:

Using that we can solve for either one of the variables in terms of the other. Contrary to the common trend, let's solve for x in terms of y.

Now the quantity that we want to maximize becomes

Now that we have a function in terms of one variable to be maximized, we can start thinking of what maximum point means. As we saw in Sec 2.1, at tha maximum point the tangent line is horizontal, which means the slope is zero. This information shows us the way to find the maximum value of a function: find where the slope is zero, and evaluate the function at that point.

Use again the slope-predictor formula to find the slope:

Solve for a when m(a)=0.

Since the maximum value occurs at y=25, plug in that value back into .