Math Analysis Test on Sections 6.3 to 6.4

Front

What is the formula for exponential decay?

Back

The formula for exponential decay is given by: $$N(t) = N_0 e^{-kt}$$ where N(t) is the quantity at time t, N_0 is the initial quantity, k is the decay constant, and e is the base of the natural logarithm.

Front

How do you approximate the value of an expression using a calculator?

Back

To approximate the value of an expression like $$\frac{(0.4e^{3\pi})}{\pi e^4}$$, input the expression into a scientific calculator and round the result to the desired number of decimal places.

Front

What is the value of $$e$$ (Euler's number)?

Back

The value of $$e$$ is approximately 2.71828.

Front

What is the significance of the constant $$\pi$$ in mathematics?

Back

The constant $$\pi$$ represents the ratio of the circumference of a circle to its diameter, approximately equal to 3.14159.

Front

What does it mean for a function to be exponential?

Back

A function is exponential if it can be expressed in the form $$f(x) = a b^x$$ where a is a constant, b is a positive real number, and x is the variable.

Front

What is the derivative of the function $$f(x) = e^x$$?

Back

The derivative of the function $$f(x) = e^x$$ is $$f'(x) = e^x$$.

Front

How do you solve for $$x$$ in the equation $$e^x = a$$?

Back

To solve for $$x$$ in the equation $$e^x = a$$, take the natural logarithm of both sides: $$x = \ln(a)$$.

Front

What is the natural logarithm?

Back

The natural logarithm, denoted as $$\ln(x)$$, is the logarithm to the base $$e$$.

Front

What is the relationship between exponential functions and logarithmic functions?

Back

Exponential functions and logarithmic functions are inverses of each other. If $$y = b^x$$, then $$x = \log_b(y)$$.

Front

What is the formula for continuous compound interest?

Back

The formula for continuous compound interest is given by: $$A = Pe^{rt}$$ where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and t is the time in years.