Calculus Core Concept Cheat Sheet

Calculus is the mathematical study of continuous change. It is primarily concerned with two major concepts: Differential Calculus (the study of rates of change and slopes of curves) and Integral Calculus (the study of accumulation and areas under curves). This cheat sheet summarizes the essential rules, formulas, and concepts required to master the fundamentals of Calculus.

1. Limits and Their Properties

A limit is the value that a function f(x) approaches as the input x approaches a certain value. Limits are essential for defining derivatives and integrals.

Basic Limit Laws

Assume lim_xa f(x) exists and lim_xa g(x) exists.

• Sum Law: lim_xa [f(x) + g(x)] = lim_xa f(x) + lim_xa g(x)
• Difference Law: lim_xa [f(x) - g(x)] = lim_xa f(x) - lim_xa g(x)
• Constant Multiple Law: lim_xa [c f(x)] = c lim_xa f(x)
• Product Law: lim_xa [f(x) g(x)] = lim_xa f(x) lim_xa g(x)
• Quotient Law: lim_xa [f(x) / g(x)] = lim_xa f(x) / lim_xa g(x) (provided lim_xa g(x) 0)

Important Limits

2. Derivatives

The derivative of a function f at a number a, denoted by f'(a), is defined as the limit of the difference quotient.

The Definition of the Derivative

Basic Differentiation Rules

Let c and n be real numbers.

• Constant Rule: d/dx [c] = 0
• Power Rule: d/dx [xⁿ] = n x^n-1
• Constant Multiple Rule: d/dx [c f(x)] = c f'(x)
• Sum and Difference Rules: d/dx [f(x) g(x)] = f'(x) g'(x)

Product and Quotient Rules

The Chain Rule

Used to differentiate composite functions: y = f(u) where u = g(x).

Derivatives of Common Functions

• d/dx [sin x] = cos x
• d/dx [cos x] = -sin x
• d/dx [tan x] = sec² x
• d/dx [sec x] = sec x tan x
• d/dx [e^x] = e^x
• d/dx [a^x] = a^x ln(a)
• d/dx [ln x] = 1/x
• d/dx [log_a x] = 1 / (x ln a)
• d/dx [sin^-1 x] = 1 / √(1 - x²)

3. Applications of Derivatives

Related Rates

Used to calculate the rate of change of one quantity based on the rate of change of a related quantity.

1. Identify the variables and constants.
2. Write an equation relating the variables.
3. Differentiate both sides with respect to time t.
4. Substitute known values and solve for the unknown rate.

Maximum and Minimum Values

Critical Number: A number c in the domain of f such that f'(c) = 0 or f'(c) does not exist.

The First Derivative Test:

• If f' changes from positive to negative at c, then f has a local maximum at c.
• If f' changes from negative to positive at c, then f has a local minimum at c.

The Second Derivative Test: Suppose f'' is continuous near c.

• If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c.
• If f'(c) = 0 and f''(c) < 0, then f has a local maximum at c.

4. Integrals

Integration is the process of finding the antiderivative of a function. It is used to calculate areas, volumes, and accumulated change.

Indefinite Integrals

Where F'(x) = f(x) and C is the constant of integration.

Basic Integration Formulas

• ∫ a dx = ax + C
• ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1)
• ∫ 1/x dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = (a^x / ln a) + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C

The Fundamental Theorem of Calculus

Connects differentiation and integration.

Part 1: If f is continuous on [a, b], then the function g defined by:

is continuous on [a, b] and differentiable on (a, b), and g'(x) = f(x).

Part 2 (Evaluation Theorem): If f is continuous on [a, b], then:

Where F is any antiderivative of f.

5. Techniques of Integration

U-Substitution

Used to reverse the chain rule. Let u = g(x). Then du = g'(x) dx.

Integration by Parts

Derived from the product rule. ∫ u dv = uv - ∫ v du.

LIPET Rule: Choose u in this order:

1. Logarithmic functions
2. Inverse trig functions
3. Polynomial (algebraic) functions
4. Exponential functions
5. Trig functions

6. Applications of Integration

Area Between Curves

The area A of the region bounded by the curves y = f(x) and y = g(x) from x = a to x = b, where f(x) g(x):

Volumes of Revolution

Disk Method

About the x-axis (horizontal).

Washer Method

About the x-axis with a hole.

Shell Method

About the y-axis (vertical).

Arc Length

Length of a curve y = f(x) from a to b.

Areas in Polar Coordinates

Area enclosed by r = f(θ) from α to β.