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Course Topics
| CHAPTER 1 Functions and Limits |
- 1.4 The Tangent and Velocity Problems
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Secant Lines, Tangent Lines, Average Velocity and Instantaneous Velocity
- 1.5 The Limit of a Function
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The limit of f(x) as x approaches a, one-sided limits, infinite limits, vertical asymptotes
- 1.6 Calculating Limits Using the Limit Laws
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Limit Laws (sum, difference, constant multiple, product, quotient, power, root), Direct Substitution Property, Squeeze Theorem.
- 1.8 Continuity
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Continuous at a (from the left, from the right), Discontinuities (removable, jump, and infinite), continuous on an interval, Intermediate Value Theorem.
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| CHAPTER 2 Derivatives |
- 2.1 Derivatives and Rates of Change
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Limit definition of the derivative of f(x) at a
- 2.2 The Derivative as a Function
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Limit definition of the derivative of f(x), differentiable at a, higher derivatives (second, third, etc.)
- 2.3 Differentiation Formulas
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Derivative of a constant, derivative of xn, Constant Multiple Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule
- 2.4 Derivatives of Trigonometric Functions
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Derivatives of sine, cosine, tangent, secant, cosecant and cotangent. Evaluating limits of the form sin(Ax)/(Bx) as x goes to 0.
- 2.5 The Chain Rule
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Computing the derivative of f(g(x))
- 2.6 Implicit Differentiation
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Computing slopes of curves defined implicitly by the equation f(x,y)=0
- 2.7 Rates of Change in the Natural and Social Sciences
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Understanding the motion of an object that travels along a straight line, computing net and total distance traveled
- 2.8 Related Rates
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Define your functions, draw and label a diagram, find an equation relating functions, differentiate both sides of this equation, plug in specific values to solve for unknown rate of change.
- 2.9 Linear Approximations and Differentials
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Using the tangent line and/or differentials to approximate values of f(x) for x near the point of tangency
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| CHAPTER 3 Applications of Differentiation |
- 3.1 Maximum and Minimum Values
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Extreme values (absolute maximum, absolute minimum), local maximum, local minimum, Extreme Value Theorem, Fermat's Theorem, critical numbers, finding the extreme values of a continuous function on a closed interval
- 3.2 The Mean Value Theorem
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Rolle's Theorem, Mean Value Theorem
- 3.3 How Derivatives Affect the Shape of a Graph
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Increasing/Decreasing Test, finding local maximums and/or minimums using First Derivative Test and Second Derivative Test, concave up, concave down, Concavity Test, inflection points.
- 3.4 Limits at Infinity; Horizontal Asymptotes
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Understanding the long term behavior of a function, what happens to f(x) as x becomes an arbitrarily large positive or negative number.
- 3.5 Summary of Curve Sketching
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Find domain of f(x), compute x and y-intercepts, determine any symmetry properties (symmetric about origin, about y-axis, periodic), find asymptotes (vertical, horizontal, slant), determine intervals of increase/decrease and use to find local max/min, determine intervals of concavity and use to find inflection points.
- 3.7 Optimization Problems
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Draw and label diagram, define all notation, determine the function that is to be optimized (if this function has more than one variable, use constraints to write function in terms of one variable), find the domain of your function, optimize using the Closed Interval Method (Section 3.1), First/Second Derivative Test (Section 3.3) or First Derivative Test for Absolute Extreme Values.
- 3.9 Antiderivatives
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Find a function whose derivative is f(x), arbitrary constant C, initial value problems.
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| CHAPTER 4 Integrals |
- 4.1 Areas and Distances
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The Area Problem, the Distance Problem, sample points, sigma notation.
- 4.2 The Definite Integral
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Left and Right Riemann Sums, Midpoint Rule, Limit definition of a definite integral, integrand, limits of integration, properties of definite integrals.
- 4.3 The Fundamental Theorem of Calculus
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Use FTOC Part 1 to compute the rate of change of indefinite integrals where the upper limit is a function of x. Use FTOC Part 2 to evaluate definite integrals using antiderivatives.
- 4.4 Indefinite Integrals and the Net Change Theorem
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Indefinite integral as most general antiderivative, table of indefinite integrals, Net Change Theorem.
- 4.5 The Substitution Rule
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Computing indefinite and definite integrals using the Substitution Rule, changing limits of integration, integrals of even and odd functions
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| CHAPTER 5 Applications of Integration |
- 5.1 Areas Between Curves
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Computing the area bounded by two or more curves.
- 5.2 Volumes
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Computing the volume of solids generated by rotation using the Disk and/or Washer method.
- 5.3 Volumes by Cylindrical Shells
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Computing the volume of solids generated by rotation using the Shell method.
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