Calculus 101

Calculus

Definition

• Branch of mathematics that deals with the study of continuous change
• Developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century

Two Main Branches

• Differential Calculus: studies rates of change and slopes of curves
• Integral Calculus: studies accumulation of quantities

Key Concepts

• Limits: the value that a function approaches as the input gets arbitrarily close to a certain point
• Derivatives: measures the rate of change of a function with respect to its input
  • Geometric Interpretation: the slope of the tangent line to the graph of the function at a point
  • Physical Interpretation: the rate at which a quantity changes over time
• Integrals: measures the accumulation of a function over a given interval
  • Definite Integral: the area between a function and the x-axis over a specified interval
  • Indefinite Integral: the antiderivative of a function, which can be used to compute definite integrals

Rules and Theorems

• Power Rule: if f(x) = x^n, then f'(x) = nx^(n-1)
• Product Rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Chain Rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)
• Fundamental Theorem of Calculus: relates the derivative of an antiderivative to the original function

Applications

• Optimization: finding the maximum or minimum of a function
• Physics: modeling the motion of objects, including the acceleration and velocity of particles and the curvature of space-time
• Economics: modeling the behavior of economic systems, including the optimization of profit and the behavior of markets

Calculus

Definition and History

• Calculus is a branch of mathematics that studies continuous change
• Developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century

Two Main Branches

• Differential Calculus: studies rates of change and slopes of curves
• Integral Calculus: studies accumulation of quantities

Key Concepts in Differential Calculus

• Limits: the value that a function approaches as the input gets arbitrarily close to a certain point
• Derivatives: measure the rate of change of a function with respect to its input
• Geometric Interpretation: the slope of the tangent line to the graph of the function at a point
• Physical Interpretation: the rate at which a quantity changes over time

Key Concepts in Integral Calculus

• Integrals: measure the accumulation of a function over a given interval
• Definite Integral: the area between a function and the x-axis over a specified interval
• Indefinite Integral: the antiderivative of a function, which can be used to compute definite integrals

Rules and Theorems in Differential Calculus

• Power Rule: if f(x) = x^n, then f'(x) = nx^(n-1)
• Product Rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Chain Rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Fundamental Theorem of Calculus

• Relates the derivative of an antiderivative to the original function

Applications of Calculus

• Optimization: finding the maximum or minimum of a function
• Physics: modeling the motion of objects, including the acceleration and velocity of particles and the curvature of space-time
• Economics: modeling the behavior of economic systems, including the optimization of profit and the behavior of markets

Trigonometry

Introduction

• Trigonometry is the study of relationships between the sides and angles of triangles.

Angles and Triangles

• Angles are measured in degrees, radians, or gradians.
• A right triangle is a triangle with one right angle (90°).

Trigonometric Ratios

Basic Trigonometric Ratios

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• Cotangent (cot) is the ratio of the adjacent side to the opposite side.
• Secant (sec) is the ratio of the hypotenuse to the adjacent side.
• Cosecant (csc) is the ratio of the hypotenuse to the opposite side.

Trigonometric Identities

Pythagorean Identity

• sin^2(A) + cos^2(A) = 1 is a fundamental identity in trigonometry.

Sum and Difference Identities

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B) is a sum identity.
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B) is a difference identity.

Graphs of Trigonometric Functions

Sine and Cosine Waves

• A sine wave is a periodic wave with a maximum amplitude of 1 and a minimum amplitude of -1.
• A cosine wave is a periodic wave with a maximum amplitude of 1 and a minimum amplitude of -1, shifted 90° from the sine wave.

Solving Triangles

Law of Sines

• a / sin(A) = b / sin(B) = c / sin(C) is used to solve triangles.

Law of Cosines

• a^2 = b^2 + c^2 - 2bc * cos(A) is used to solve triangles and find the third side.

Applications of Trigonometry

Real-World Applications

• Navigation uses trigonometry to calculate distances and directions using triangulation.
• Physics models periodic motion, such as sound waves and light waves, using trigonometry.
• Engineering uses trigonometry to calculate stresses and strains in structures.