Natural Exponential Functions and Graphs

This lecture, part of a pre-calculus course, focuses on natural exponential functions. It begins with an introduction to the number e, explaining its properties as an irrational number and its role as a base in exponential functions. The lecture then covers natural exponential functions, their graphs, and how to handle shifts and transformations. It also explores graphing functions with absolute values and solving complex exponential expressions. The session concludes with finding intersection points of graphs and providing practice questions for further learning.

What is the main focus of Lecture 3 in the pre-calculus course?

• Trigonometric functions

• Logarithmic functions

• Natural exponential functions

• Polynomial functions

Which of the following is an irrational number discussed in the lecture?

• 2.5

• 1.618

• 2.71828

• 3.14

How is the number e derived?

• As the sum of an infinite series

• As the limit of (1 + 1/n)^n as n approaches infinity

• As the solution to a quadratic equation

• As the product of prime numbers

What is the base of a natural exponential function?

• e

• π

• 2

• 10

What happens to the graph of an exponential function as the base increases?

• It moves away from the y-axis

• It becomes a parabola

• It becomes a straight line

• It gets closer to the y-axis

In the example of a shifted exponential function, what is the effect of adding 2 to the exponent?

• Shifts the graph to the left by 2 units

• Shifts the graph down by 2 units

• Shifts the graph to the right by 2 units

• Shifts the graph up by 2 units

What is the range of the function f(x) = e^x after it is shifted down by 1 unit?

• (-∞, ∞)

• (0, ∞)

• (-1, ∞)

• (1, ∞)

How do you handle the negative part of a graph when dealing with absolute values?

• Make it positive

• Reflect it over the x-axis

• Ignore it

• Shift it up by 1 unit

What is the result of the expression f(x+2)/f(x-1) + 2f(2)f(x+1)/f(x) when simplified?

• 2e^3

• e^2

• 3e^3

• e^3

What is the intersection point of the graphs of f(x) = e^(-3-x) and g(x) = √(e^5)?

• (-11/2, e^(5/2))

• (11/2, e^(5/2))

• (-11/2, e^5)

• (11/2, e^5)