FFR - Estimating Areas

A new branch of Calculus

For the longest time, there were two separate branches of calculus. The first was derivative calculus, where mathematicians estimated slopes of tangent lines using limits, and there was integral calculus, where mathematicians used limits to estimate areas under curves. Whichever branch you choose, the foundation of Calculus is based on limits.

Using integral calculus, we can set up the area under the curve as small, differential areas (differential just means really really thin). For the sake of simplicity, let’s make these areas rectangular. That way, we only have to worry about length and width. We can also use other shapes for estimating areas, but their calculations are more complicated.

This is going to be a relatively short section, but with lots of pictures to develop a conceptual framework.

Example Problem 6.4.1

Using the trapezoids method, estimate the area under the curve of $y=\sin (\pi x)$ from x = 0 to x = 2, with n, the number of trapezoids, equal to 4.

Chapter 6 Problems

1. You open up a business using $10 000 of your hard-earned money. You think that the rate of return on your investment (as a decimal) can be modeled by the equation $r=e^{-t}+0.07$. What is the rate of increase of the assets in your business after 3 years? Use units of dollars per year.

   Clarification: Use the asset model $A\left(t\right)=10000e^{t(e^{-t}+0.07)}$. Here the given r(t) is inserted into the exponent t r(t); it is not an instantaneous proportional growth rate.

2. You need $50 000 per year to live comfortably when you retire. Your bank offers a high interest savings account that gives you a compounding interest of 5% per year. How much money do you need to invest now if you want to live comfortably off the interest during your retirement 30 years from now ?

   Clarification: Assume continuous compounding. The retirement fund is chosen so that interest at 5% supplies $50,000 per year without using the principal.

2. ?

3. You want to maximize revenues for your company by changing the price. Your sales follow the function $S=8-3\ln P$, where S is the quantities sold (in 10 000 units) and P is the price. You want to keep the price of your product between $1 and $12. ​
   a) What is the optimum price point?​
   b) How many units will you sell at this price?
4. You want to maximize profits by changing the quantity of laptops your company produces. Each laptop brings in $1000 of revenue. The total cost of producing laptops per year is represented by $C=\frac{Q^3}{\mathrm{10,000}}$, where C is the cost and Q is the quantity of laptops produced.​
   a) What is the optimum number of laptops your company should make for profit maximization?​
   b) What is the annual profit at that quantity?

5. As the size of your company increases, its percentage growth decreases. However, your company is adding more employees per year than ever before! Explain these concepts using variables, derivatives, and related rates.
6. What is the absolute maximum value of the graph of $y=10x-x^2$

7. For the equation $y=3\ln x-\sqrt{x}$, ​
   a) At which point does its absolute maximum value occur?​
   b) What is the maximum value?
8. As you pump air into a spherical balloon at a rate of 3cm³ per second, its radius increases to 2 cm. What is the rate of change of its surface area?

9. You have an inflatable cube that constantly increases in size at 1cm per second. What is the rate of change in its volume when one of its sides is 4 cm?
10. Your friend gets distracted while holding up your ladder and you feel yourself starting to fall. Assuming gravity causes the vertical displacement from your original position to be defined as $x=-3t^2$, at what rate is the angle between the ladder and the ground changing after 0.5s?​
    

    Clarification: Treat the ladder as rigid, with its foot sliding on horizontal ground. Use the source diagram’s initial height 5 m and base distance 3 m; the top height is 5−3t² metres.