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notes - MAT1300 Notes By Eric Hua Contents Chapter 0 A...

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MAT1300 – Notes — By Eric Hua Contents Chapter 0. A Precalculus Review 3 Chapter 1. Functions, Graphs, and Limits 6 1.1 The Cartesian Plane and Distance Formula . . 6 1.2 Graphs of Equation . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Lines in the plane and slope . . . . . . . . . . . . . . . . 8 1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 2. Differentiation 14 2.1 The derivative and the slope of a graph . . . . . . 14 2.2 Some Rules for differentiation . . . . . . . . . . . . . . . 16 2.3 Rates of Change: Velocity and Marginals . . . . 18 2.4 The product and Quotient rules . . . . . . . . . . . . 20 2.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Higher derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Implicit differentiation . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3. Applications of the Derivative 24 3.1 Increasing and Decreasing Functions . . . . . . . . . 24 3.2 Extrema and the First-Derivative Test . . . . . . . 25 3.3 Concavity and Second-Derivative Test . . . . . . . 26 3.4 Optimization Problems . . . . . . . . . . . . . . . . . . . . . 28 3.5 Business and Economics Applications . . . . . . . . 30 3.6 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.7 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1
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Chapter 4. Exponential and Logarithmic functions 34 4.1 Exponential functions . . . . . . . . . . . . . . . . . . . . . 34 4.2 Natural Exponential function . . . . . . . . . . . . . . . 35 4.3 Derivatives of Exponential Functions . . . . . . . . 36 4.4 Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Derivatives of Logarithmic Functions . . . . . . . . 37 4.6 Exponential Growth and Decay . . . . . . . . . . . . . 38 Chapter 5. Integration and its Applications 40 5.1 Antiderivatives and Indefinite Integral . . . . . . . 40 5.2 Integration by Substitution and the General Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.3 Integral of Exponential and Logarithm . . . . . . . 42 5.4 Area and Fundamental Theorem of calculus . . 44 5.5 Area between two Graphs . . . . . . . . . . . . . . . . . . 46 Chapter 6. Techniques of Integration 49 6.1 Integration By Parts and Present Value . . . . . . 49 6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 7. Functions of Several Variables 52 7.1 The 3D Coordinate System . . . . . . . . . . . . . . . . . 52 7.2 Surfaces in Space (only first 2 pages) . . . . . . . . 52 7.3 Functions of Several Variables . . . . . . . . . . . . . . . 53 7.4 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.5 Extrema of Functions of Two Variables . . . . . . 55 2
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Chapter 0. A Precalculus Review 1. Real numbers and intervals Interval Notation Set Notation [ a, b ] { x R : a x b } ( a, b ) { x R : a < x < b } [ a, b ) { x R : a x < b } ( a, b ] { x R : a < x b } ( a, + ) { x R : x > a } [ a, + ) { x R : x a } ( -∞ , b ) { x R : x < b } ( -∞ , b ] { x R : x b } ( -∞ , + ) R 2. Solving inequalities Example 1 Solve the inequality - 2 x - 3 ≤ - 13 . Solution: We have - 2 x - 3 ≤ - 13 ⇒ - 2 x ≤ - 13 + 3 ⇒ - 2 x ≤ - 10 . The next step would be to divide both sides by - 2. Since - 2 < 0, the sense of the inequality is inverted, and so - 2 x ≤ - 10 x - 10 - 2 x ≥ - 5 . Example 2 Solve the inequality x 2 + 2 x - 35 < 0 . Solution: Observe that x 2 + 2 x - 35 = ( x - 5)( x + 7), which vanishes when x = - 7 or when x = 5. Now we construct the table: x ( -∞ , - 7) ( - 7 , 5) (5 , + ) x + 7 - + + x - 5 - - + ( x + 7)( x - 5) + - + 3
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On the last row, the sign of the product ( x + 7)( x - 5) is determined by the sign of each of the factors x + 7 and x - 5. From the sign diagram above we see that { x R : x 2 + 2 x - 35 < 0 } = ( - 7 , 5) . Notice that we exclude both x = - 7 and x = 5 in the set, as ( x + 7)( x - 5) vanishes there. 3. Absolute Values Definition 1 Let x R . The absolute value of x —denoted by | x | —is defined by | x | = ( - x if x < 0, x if x 0.
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