Calculus Chapter 1: Functions, Radian Measure, and Trigonometric Functions

Functions

Definition and Properties of Functions

In calculus, a function is a rule that assigns a single output value to each input value. Functions are fundamental objects in mathematics, especially in calculus, where they are used to describe relationships between varying quantities.

• Notation: If the input is x and the function is f, the output is written as f(x).

• Graph: The graph of a function f is the set of all points (x, y) in the Cartesian plane such that y = f(x).

• Independent and Dependent Variables: The input variable is called the independent variable, and the output is the dependent variable.

• Domain Restrictions: Some functions are only defined for certain input values.

• Piecewise Functions: Some functions are defined by different formulas for different input values.

• Function Notation: It is customary to refer to f(x) rather than just f to avoid ambiguity.

Example: If f(x) = x^2 - x + 1, then f(2) = 2^2 - 2 + 1 = 3.

Inverse Functions

An inverse function reverses the effect of the original function, mapping outputs back to their corresponding inputs. Not all functions have inverses; a function must be one-to-one (injective) for its inverse to exist.

• Notation: The inverse of f is denoted f^{-1}.

• Example: The exponential function a^x and the logarithmic function \log_a x are inverses: if y = a^x, then x = \log_a y.

• Restriction for Inverses: If a function is not one-to-one, its domain may be restricted to ensure the existence of an inverse. For example, \sqrt{x} is the inverse of x^2 restricted to x \geq 0.

Example: The inverse of f(x) = x^2 (for x \geq 0) is f^{-1}(x) = \sqrt{x}.

Radian Measure

Definition and Conversion

Angles can be measured in radians, a natural unit based on the arc length of a circle. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius.

• Arc Length Formula: $s = r\theta$

• Conversion: $\theta = \frac{\pi \alpha}{180}$, where \alpha is in degrees and \theta is in radians.

• Key Values:

  • 360° = 2\pi radians

  • 180° = \pi radians

  • 90° = \frac{\pi}{2} radians

  • 60° = \frac{\pi}{3} radians

  • 45° = \frac{\pi}{4} radians

  • 30° = \frac{\pi}{6} radians

Example: Convert 60° to radians: $60° = \frac{\pi}{3}$ radians.

Area of a Sector

The area of a sector of a circle with radius r and angle \theta (in radians) is given by:

• Formula: $A = \frac{1}{2} r^2 \theta$

Example: For a sector with r = 4 and \theta = \frac{1}{2} radians, $A = \frac{1}{2} \times 16 \times \frac{1}{2} = 4$.

Applications: Goat in a Square Field

Consider a goat tied to the corner of a square field of side 100 m with a rope of length r. The goat can graze an area equal to a quarter circle of radius r plus a region within the square. If the goat is to reach exactly half the grass in the field, we set the grazed area equal to half the area of the square.

• Area of Square: $A_{\text{square}} = 100^2 = 10,000$ m2

• Area Grazed: $A_{\text{grazed}} = \frac{1}{4} \pi r^2$

• Set Equal: $\frac{1}{4} \pi r^2 = 5,000$

• Solve for r: $r = 100 \sqrt{\frac{2}{\pi}}$

Additional info: This classic problem demonstrates the use of radian measure and area of a sector in a real-world context.

Trigonometric Functions

Definition and Geometric Interpretation

Trigonometric functions relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, they are defined as follows for an angle \theta:

• cosine: $\cos \theta = \frac{x}{r}$

• sine: $\sin \theta = \frac{y}{r}$

• tangent: $\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}$

On the unit circle (r = 1), the coordinates of a point are (\cos \theta, \sin \theta).

Other Trigonometric Functions

• cotangent: $\cot \theta = \frac{\cos \theta}{\sin \theta}$

• secant: $\sec \theta = \frac{1}{\cos \theta}$

• cosecant: $\csc \theta = \frac{1}{\sin \theta}$

Key Identities and Properties

• Pythagorean Identity: $\cos^2 \theta + \sin^2 \theta = 1$

• Other Identities:

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

• Angle Addition:

  • $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

  • $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Double Angle:

  • $\cos 2A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

  • $\sin 2A = 2\sin A \cos A$

• Even-Odd Properties:

  • $\cos(-\theta) = \cos \theta$

  • $\sin(-\theta) = -\sin \theta$

Solving Trigonometric Equations

Trigonometric equations can be solved by expressing them in terms of sine, cosine, or tangent and using the identities above. The general solutions are:

• $\cos \theta = \cos \alpha \implies \theta = \pm \alpha + 2n\pi$

• $\sin \theta = \sin \alpha \implies \theta = \alpha + 2n\pi$ or $\theta = \pi - \alpha + 2n\pi$

• $\tan \theta = \tan \alpha \implies \theta = \alpha + n\pi$

Example: Solve $\cos 2\theta = \sin \theta$. Use identities to rewrite and solve for \theta.

Standard Angles and Triangles

Values of trigonometric functions for standard angles (such as $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$) are derived from special right triangles and are frequently used in calculus.

Summary Table: Trigonometric Function Values for Standard Angles

Angle (degrees)	Angle (radians)	sin	cos	tan
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

Additional info: Mastery of these values and identities is essential for calculus, especially for differentiation and integration involving trigonometric functions.