Lesson 1: Review
MA16010 — Applied Calculus I
Review: Exponential Rules
Exponential Properties
(1) \(x^a x^b = x^{a+b}\)
(2) \(\frac{x^a}{x^b} = x^{a-b}\)
(3) \((x^a)^b = x^{ab}\)
(4) \(x^1 = x\)
(5) \(x^0 = 1\)
(6) \(x^{-1} = \frac{1}{x}\)
These exponential rules are fundamental for working with powers and will be used extensively throughout calculus. Remember that the base \(x\) must be nonzero for rules involving division and negative exponents.
Review: Logarithmic Rules
Logarithmic Properties
(1) \(\ln 1 = 0\)
(2) \(\ln(e^x) = x\)
(3) \(e^{\ln x} = x\)
(4) \(\ln(xy) = \ln x + \ln y\)
(5) \(\ln\left(\frac{x}{y}\right) = \ln x - \ln y\)
(6) \(\ln(x^m) = m \ln x\)
The natural logarithm function \(\ln\) is the inverse of the exponential function \(e^x\). Properties (2) and (3) reflect this inverse relationship. The logarithmic properties in (4), (5), and (6) are essential for simplifying logarithmic expressions and solving logarithmic equations.
Review: Trigonometry
Trigonometric Function Values at Special Angles
The following table summarizes the values of sine and cosine at commonly used angles. The angles are given in radians, where \(\pi/6\) corresponds to 30°, \(\pi/4\) to 45°, \(\pi/3\) to 60°, and \(\pi/2\) to 90°.
| Angle | \(\theta = 0\) | \(\theta = \frac{\pi}{6}\) | \(\theta = \frac{\pi}{4}\) | \(\theta = \frac{\pi}{3}\) | \(\theta = \frac{\pi}{2}\) |
|---|---|---|---|---|---|
| \(\sin \theta\) | \(0 = \frac{0}{2}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{4}}{2} = 1\) |
| \(\cos \theta\) | \(1 = \frac{\sqrt{4}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(\frac{0}{2} = 0\) |
Notice the pattern in the table: the numerators for sine increase as \(\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}\) (all divided by 2), while the cosine values follow the reverse pattern. This symmetry can help with memorization.
To find trigonometric values in the other quadrants, we use the ASTC diagram (which stands for "All Students Take Calculus") along with reference angles. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.
ASTC Diagram: Quadrant Sign Convention
A mnemonic device for remembering which trigonometric functions are positive in each quadrant:
Quadrant I (A): All functions positive
Quadrant II (S): Sine positive
Quadrant III (T): Tangent positive
Quadrant IV (C): Cosine positive
Additional Trigonometric Functions
Reciprocal and Quotient Trigonometric Functions
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
\(\csc \theta = \frac{1}{\sin \theta}\)
\(\sec \theta = \frac{1}{\cos \theta}\)
These four additional trigonometric functions are defined in terms of sine and cosine. The tangent and cotangent are quotient functions, while the cosecant and secant are reciprocal functions. Be mindful of where these functions are undefined: tangent and secant are undefined when \(\cos \theta = 0\), and cotangent and cosecant are undefined when \(\sin \theta = 0\).
Key Reminders
When working with trigonometric functions in calculus, remember to work in radians rather than degrees unless otherwise specified. The derivatives and integrals of trigonometric functions are derived using radian measure.