The average rate change approaches the instanta-neous rate change

Definition 6 (One-to-one function) A function f is one-to-one if, for any a and b in the domain of f such that a ̸= b, then f(a) ̸= f(b). In other words, each output yields a unique input.

• A function is one-to-one if any horizontal line cuts the function at one or fewer points. (This is called the horizontal line test.)

Definition 8 (Polynomial) A polynomial is a function consisting only of pow-ers of the variable (usually x) multiplied by constant coefficients.

Definition 9 (Rational Function) A rational function is of the form f(x) = P(x)
Q(x), where P(x) and Q(x) are polynomials. Note that Q(x) ̸= 0.

▷ The triangle inequality states that |a + b| ≤ |a| + |b|.

1.3 The Greatest Integer Function

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1. sin(−x) = − sin x
2. cos(−x) = cos x
3. sin(π 2− x) = cos x
4. cos(π 2− x) = sin x
5. sin(π 2+ x) = cos x

6. cos(π 2+ x) = − sin x
7. sin(π − x) = sin x
8. cos(π − x) = − cos x
9. sin(π + x) = − sin x
10. cos(π − x) = − cos x

3. cot2x + 1 = csc2x

1.4.4 Sum and Difference Formulas

5. tan(α + β) = tan α + tan β

1 − tan α tan β

1. sin2α = 1 − 2 cos(2α)
2

2. cos2α = 1 + 2 cos(2α)
2

4. c2= a2+ b2− 2ab cos C (Law of Cosines)
5. Area of triangle =1 2ab sin C

6. Area of triangle =

• sin x, cos x, csc x, and sec x all have periods of 2π.
• tan x and cot x have periods of π.

▷ If the x in sin x, cos x, etc., is multiplied by a constant b, the period is divided by that constant:

1.4.8 Inverse Trig Functions

If f(x) = sin x, then
f−1(x) = sin−1x = arcsin x, with −1 ≤ x ≤ 1
If f(x) = cos x, then
f−1(x) = cos−1x = arccos x, with −1 ≤ x ≤ 1
If f(x) = tan x, then
f−1(x) = tan−1x = arctan x, with −π 2≤ x ≤ π

3. am· an= am+n
4. am÷ an= am−n
5. (am)n= amn

6. a−m= 1
am

1. loga 1 = 0

2. loga a = 1

1.6 Parametric Functions

Definition 14 (Parametric Equations) A set of equations that define sev-eral variables (usually two) in terms of another variable.

f′(a) = lim f(a + h) − f(a)
h→0 h

If f’(a) exists, then from the above equation, we know that limx→c = f(c). So if a function is differentiable, the it is continuous. (However, the reverse is not necessarily true; a function may be continuous at a point but not be differentiable at that point.

6. There is a cusp at x = a, so there are infinitely many tangents passing through (x, f(x)).

2.1 The Chain Rule

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▷ Logarithmic Differentiation (for derivatives of exponential functions): y = f(x)
ln y = ln f(x)

Theorem 2 (The Mean Value Theorem (MVT)) If the function f(x) is

continuous on the interval [a, b] and differentiable on the interval (a, b), then

Theorem 3 (Rolle’s Theorem) If f(a) = f(b) = 0, then for some c in [a, b],

f′(c) = 0.

•x→∞f(x) = lim x→∞g(x) = ∞

then the limit in question is equal to

the latter, rewrite it as eln(00)and apply the laws of logarithms.

2.7 Estimating

2h

3 Applications of Differentiation

Definition 16 (Critical Point) A critical point is a point at x = c such that f′(c) = 0 or f′(c) is undefined. To determine the critical points of a function, find its derivative, determine which values of x make it undefined, and solve for f′(x) = 0 for the remaining values.

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