Applications

This chapter introduces application tasks related to root, exponential and logarithmic functions

Example 1

The number of visitors on a website grows by an average of 1,7 percent per week.

a) What percentage of the number of visitors grows per year?

b) After how many years has the number of visitors increased 100 times?

c) If the site has 1200 visitors at the beginning of the review period, what is the growth rate of the number of visitors after two years?

We denote the number of visitors at the beginning of the reference period with the letter a.

a) After one year, the number of visitors is

The number of visitors has increased 2,4 times. The number of visitors has increased by 140 percent

b)

The number of visitors has increased 100 times after 273 weeks, that is after about five years and three months.

c) Differentiate the function of the expression for the number of visitors

In two years, the number of visitors will increase by about 125 visitors per week.

Example 2

The length of the sides of an isosceles triangle is 3. Find the maximum area of the triangle.

Let the letter x denote the half of the base of the triangle. The variable x can vary between [0,3] We solve the height h of the triangle and form the expression for the area of the triangle.

To find the maximum value of the area, a derivative of the function A(x) is formed and the zeros of the derivative are determined. The derivative formula of the product function could be used here

(fg) ’= f’g + g’f but another solution is presented here. We place x inside the square root, after which it is enough to look at the expression inside the square root

Now the function A(x) gets its maximum value when the expression inside the root gets its maximum value. We denote by g(x) the expression inside the root and solve the zeros of its derivative

x cannot be negative, so the negative root can be ignored. The maximum value of the function A(x) is reached at the zero point of the derivative or at the endpoints of the interval.

The maximum value of the area of a triangle is 9/2. Let's draw graph of the area function