Continuous Compounding and Exponential Functions

The video tutorial covers the concept of base e and natural logarithms, explaining their properties and significance in mathematics. It discusses exponential growth and decay, and how these functions relate to their inverses, the natural logarithms. The tutorial also demonstrates how to convert between exponential and logarithmic forms, solve equations and inequalities using natural logs, and apply these concepts to continuous compounding interest calculations.

What is the approximate value of the natural base 'e'?

• 3.14159

• 2.71828

• 1.61803

• 1.41421

Which function represents exponential growth when the exponent is positive?

• f(x) = x^e

• f(x) = e^(-x)

• f(x) = ln(x)

• f(x) = e^x

How is the natural logarithm of a number 'x' commonly written?

• e^x

• log_e(x)

• ln(x)

• log(x)

Convert the exponential equation e^7 = x into logarithmic form.

• ln(7) = x

• log(x) = 7

• ln(x) = 7

• log_e(x) = 7

What is the result of e raised to the power of ln(x)?

• e^x

• 1

• ln(x)

• x

Solve for x: 3 * e^(-2x) = 6.

• x = ln(1)/-2

• x = ln(6)/-2

• x = ln(2)/-2

• x = ln(3)/-2

What is the formula for continuous compounding interest?

• A = P * e^(n * t)

• A = P * (1 + rt)

• A = P * e^(R * T)

• A = P * (1 + r/n)^(nt)

If $700 is invested at 3% interest compounded continuously, what is the balance after 8 years?

• $950.00

• $900.00

• $889.87

• $850.00

How long will it take for an account to reach at least $1,200 if $700 is invested at 3% interest compounded continuously?

• 12 years

• 20 years

• 18 years

• 15 years

What principal amount is needed to achieve a balance of $1,500 after 12 years at 3% interest compounded continuously?

• $1,550

• $1,465

• $1,500

• $1,400