Lesson 20MA 152, Sections 4.1 and 4.2
We have discussed powers where the exponents are integers or rational numbers. There also exists powers such as . You can approximate powers on your calculator using the power key. On most one-liner scientific calculators, the power key looks like
Enter the base into the calculator first, press the power key, enter the exponent, and press enter or equal.
Ex 1:Approximate the following powers to 4 decimal places.
If , the function defines a function. Since x can be any number, the domain is the set of all real numbers. Since the base b is positive, the value of y will always be positive, so the range is .
Exponential Function: An exponential function is defined by the equation
Ex 1:Graph each exponential function.
Generalizations about graphs of exponential functions of the form y = bx:
- The y-intercept will always be (0, 1) because any base to the 0 power is 1.
- If the base is greater than 1, the graph is increasing. If the base is between 0 and 1, the graph is decreasing.
- The graph lies above the x-axis, because the range is only positive values.
- The x-axis is an asymptote of the graph. An asymptote is a line that the graph approaches but never touches.
- The point (1, b) is on the graph.
Ex 2:Determine which of these graphs could be an exponential function of the form .
Ex 3:Approximate a possible equation for graph (a) above. (Remember, the point
(1, b) will be on the graph.
Compound Interest: If a bank account earns interest on a regular basis and the interest earned is added to the total before the next interest is figured, this is called compound interest. Some banks figure interest monthly, quarterly, semi-annually, etc. The formula used to figure compound interest is , where A is the final amount in the account, P is the initial deposit or amount, r is the interest rate earned (as a decimal), k is the number of times in a year that interest is figured, and t is time in years. As k gets very large, so that interest is compounded constantly, we say interest is compounded continuously. What happens to the formula when k becomes very large? It becomes the following formula, . The A, P, r, and t represent the same as in the regular compound interest formula. What is the number e? If the quantity is evaluated for very large values of x, it becomes approximately 2.71828182845904... . As x become infinity, this number is called e.
You will notice that your calculator has above a key marked ln on your scientific calculator. The base e is approximately 2.71828... .
To find these powers:
- Enter the power in your calculator
- Because the e power is above the ln key, you must press the key first and then the key.
- The value is approximately the power.
Ex 4:Approximate each power to 4 decimal places.
Compound Interest Formula: If P dollars are deposited in an account earning interest at an annual rate r, compounded continuously, the amount A after t years is given by the formula .
Ex 5:Jim deposited $4500 at 5.4% annual interest rate in an account compounded continuously. How much is in Jim's account at the end of 5 years? (Assume he made no more deposits into the account.)
Ex 6:Lily's parents deposited an amount in her account on her day of birth. The account earned 6% annual interest and on her 18th birthday the account was $40,000. How much was the initial deposit by her parents?
Other application problems use formulas that contain an exponential expression as part of the formula, some with base e, others with different bases. Several of these formulas are found in lesson 4.2 in your textbook. We will use some of them in these problems.
The half-life of an element is the amount of time necessary for the element to decay to half the original amount. Uranium is an example of an element that has a half-life. The half-life of radium is approximately 1600 years. The formula used to find the amount of radioactive material present at time t, where A0 is the initial amount present (at t = 0), and h = half-life of the element is .
Ex 7:Tritium, a radioactive isotope of hydrogen, has a half-life of 12.4 years. If an initial sample has 50 grams, how much will remain after 100 years?
Another formula represents the population growth. The formula is where P is the final population, P0 is the initial population at time 0, t is time in years, and k = b - d (b is birth rate and d is death rate).
Ex 8:The population of a city is 45,000 in 2008. The birth rate is 11 per 1000 and the death rate is 8 per 1000. What will be the population in 2018?
The intensity of light I (in lumens) at a distance of x meters below the surface of water is represented by , where I0 is the intensity of light above the water and k is a constant that depends on the clarity of the water.
Ex 9:At the center of a certain lake, the intensity of light above the water is 10 lumens and the value of k is 0.8. Find the intensity of light at 3 meters below the surface.
Ex 10:The population of EagleRiver is growing exponentially according to the model, , where t is years from the present date. Find the population in 6 years.
Ex 11:For one individual the percent of alcohol absorbed into the bloodstream after drinking two shots of whiskey is given by , where t is in minutes. Find the percent of alcohol in the bloodstream after ½ hour.
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