Exponential Functions Lecture PDF

Exponential functions Conor McCoid January 7th, 2025 McMaster University Lecture objectives Main goal Learn about exponential functions Concepts: ˆ what is an exponential function ˆ what is special about exponential functions ˆ what are some applications of exponential f...

Exponential functions Conor McCoid January 7th, 2025 McMaster University Lecture objectives Main goal Learn about exponential functions Concepts: ˆ what is an exponential function ˆ what is special about exponential functions ˆ what are some applications of exponential functions Textbook sections 4.1 1 Theory Definition of exponential growth Exponential growth A function grows exponentially if its growth depends on its value 2 Definition of an exponential function Exponential function For a constant b > 0, the function f (x) = b x is an exponential function 3 Properties of an exponential function For b > 0: ˆ b x is defined for all −∞ < x < ∞ ˆ b x is continuous (there are no jumps or singularities) ˆ b x > 0 for all x ˆ b x intercepts the y -axis at (0, 1) 4 Limits of an exponential function For b > 1 (representing exponential growth): ˆ limx→−∞ b x = 0 ˆ limx→∞ b x = ∞ For 0 < b < 1 (representing exponential decay): ˆ limx→−∞ b x = ∞ ˆ limx→∞ b x = 0 5 Rules for exponential functions Equality: for b ̸= 1, b x = b y if and only if x = y Product: b x b y = b x+y bx Quotient: by = b x−y Power: (b x )y = b xy Multiplication: (ab)x = ax b x
x x Division: ba = bax 6 Reflection Take a moment to ˆ breathe; ˆ discuss with a neighbour, or; ˆ come up with a question. 7 Application Future value of an investment Future value Suppose a principal P is invested at an annual interest rate r for t years to accumulate a future value B(t). If interest is compounded k times per year, then  r kt B(t) = P 1 +. (1) k If instead interest is compounded continuously, then B(t) = Pe rt. (2) 8 Continuous compounding As k, the number of times per year that interest is compounded, increases, the two formulas become closer to one another. That is,  r kt lim P 1 + = Pe rt. k→∞ k Using our rules for exponential functions, we can reduce this to  r k 1+ = er k 9 Euler’s number Euler’s number Euler’s number, denoted as e, may be defined as 1 k   lim 1 + = e ≈ 2.72. k→∞ k Euler’s number is going to be very important for us. The function e x is often referred to as the exponential function. 10 How much does the principal need to be? Suppose you want to have a certain amount of money after a certain number of years. That is, you know you want a specific value of B(T ) after some time T. Suppose r and k are fixed, then the only thing you can control is P, the initial principal. How much does this need to be so that you get to B(T )? We want to take the future value formula and isolate for P. We do this by dividing by everything that multplies P:  r kT B(T ) B(T ) = P 1 + =⇒ P =
kT k 1 + kr 11 Present value of an investment Present value Suppose an investment B will be accumulated at an annual interest rate r compounded k times per year over T years. The present value of this investment P(T ) is  r −kT P(T ) = B 1 +. (3) k If instead interest is compounded continuously, then P(T ) = Be −rT. (4) 12 Effective interest When interest is compounded multiple times per year, the nominal annual interest rate r does not reflect how much money is accumulated at year’s end. We’re interested in the effective annual interest rate, which tells us how much money is earned per year.  r k B(1) = P 1 + = P(1 + re ) k 13 Effective interest rate Effective interest rate If interest is compounded k times per year at a nominal rate r , then the effective annual interest rate re is  r k re = 1 + − 1. (5) k If instead interest is compounded continuously, then re = e r − 1. (6) 14 Example Compare linear and exponential growth Suppose you invest $1000 at a nominal annual rate of 1.2% that is compounded monthly. How much is the investment worth after two full years (24 months)? Suppose instead the bank deposits $1 into the account each month. Compare this linear growth with the exponential growth above. Additional questions: ˆ How much needs to be invested initially to earn $1000 after 24 months? ˆ What is the effective annual interest rate? ˆ What if the interest was compounded weekly? Compare effective interest rates. 15 16 17 18 19 20 Summary ˆ Functions of the form b x are called exponential functions ˆ Exponential functions can be used to model compound interest ˆ Effective interest rates differ from nominal interest rates based on the number of times interest compounds per year 21 Exercises Section 4.1: Curve sketching: 3, 4, 30 (use a graphing calculator or Wolfram Alpha) Simplify these expressions: 5, 9, 13, 17 Solve these equations: 19, 25 Apply formulas: 35, 37, 39 Word problems: 43, 47 22