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Questions and Answers
What is the main focus of calculus?
What is the main focus of calculus?
Who are the two main developers of calculus?
Who are the two main developers of calculus?
What is the geometric interpretation of a derivative?
What is the geometric interpretation of a derivative?
What is the physical interpretation of a derivative?
What is the physical interpretation of a derivative?
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What is the power rule of differentiation?
What is the power rule of differentiation?
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What is the fundamental theorem of calculus?
What is the fundamental theorem of calculus?
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What is one of the main applications of calculus in physics?
What is one of the main applications of calculus in physics?
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What is one of the main applications of calculus in economics?
What is one of the main applications of calculus in economics?
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The sine of an angle in a right triangle is always equal to the cosine of its complementary angle.
The sine of an angle in a right triangle is always equal to the cosine of its complementary angle.
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The law of cosines is only used for right triangles.
The law of cosines is only used for right triangles.
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The Pythagorean identity is sin^2(A) + cos^2(A) = 0.
The Pythagorean identity is sin^2(A) + cos^2(A) = 0.
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Trigonometry is only used in navigation and physics.
Trigonometry is only used in navigation and physics.
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The cotangent of an angle is always equal to the reciprocal of the tangent.
The cotangent of an angle is always equal to the reciprocal of the tangent.
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The graph of the sine function is a linear function.
The graph of the sine function is a linear function.
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Study Notes
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.
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Description
An introduction to calculus, a branch of mathematics that deals with the study of continuous change. Learn about its history, two main branches, and key concepts.
