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Principles of Calculus Modeling
An Interactive Approach
By Donald Kreider and Dwight Lahr
1 Modeling Discrete Data
 1.1 Introduction to the Issues

 1.2 Lines in the Plane

 1.3 Functions and Their Graphs
 Is f(x) even, odd, or neither?
 cos(x)
 1/(x^{3} + x^{5})
 x·e^{x^2}
 cos(2x) + 3
 sin(x) + x^{2}
 1.4 Defining New Functions from Old

 Find domain and range from a sketch
 Given domain and range of f(x), find domain and range of g(x) = f(1 – x)
 Given domain and range of f(x), find domain and range of g(x) = f(x^{3}) – 1
 Given domain and range of f(x), find domain and range of g(x) = 2f(2x) – 1
 find the domain and range of f(x) = –12 – 10√x
 find the domain of f(x) = 10 / (5 – √(100 – x))
 find the inverse function of f(x) = 12 + x^{(1/3)} / 2
 1.5 Trigonometric Functions

 Tricks to translate and scale sin(x)
 Graph cos(5(x – π/2))
 Graph 3sin(2x)
 Basic trig
 1.6 Exponential and Logarithm Functions

 graphs of e^{x} and e^{(.5x)}
 If e^{(2x +7)} = 5, what is x?
 Graphing ln x
 If ln(3x^{2}) = 10 what is x?
 1.7 Case Study: Modeling with Elementary Functions

2 Modeling Rates of Change
 2.1 Introduction to the Issues

 Traveling 60 miles in half an hour gets you a speeding ticket
 Derived tables for x^{2}
 2.2 The Legacy of Galileo and Newton

 You hike 2 miles in 35 minutes, what is your average velocity?
 You hike 2 miles in 35 minutes then 4 miles in 60 minutes, what is your average velocity?
 You hike 2 miles in 35 minutes then 4 miles in 60 minutes then .5 miles in 6 minutes, what is your average velocity? during which segment was your average velocity the greatest?
 2.3 Limits of Functions

 Left and righthanded limits of a function may be different or the same
 Sandwich Theorem shows the limit as x approaches 0 of sin(x)/x = 1
With G(x) a piecewise defined, bumpy function:
 limit as x approaches 1 from the right of G(x) = 0
 limit as x approaches 3 from the left of G(x) = 0
 limit as x approaches 3 of G(x) = 0; limit as x approaches 1 of G(x) is undefined
Evaluate these limits:
 f(x) = 1/(x^{2} – 16)
 limit as x approaches 7 from the right of (3x)/(x^{2} – 8x + 7)
 limit as x approaches 7 from the left of (x – 7)/(x^{2} – 8x + 7)
 2.4 Limits at Infinity

 limit as x approaches ∞ of x ·sin(x)
 limit as x approaches ∞ of sin(x)/x
 2.5 Continuity

 Left and right continuous defined
Describe continuity of:
 f(x) = 3x for x < 1 f(x) = 4x for x ≥ 1
 f(x) = 2x for x < 0, f(x) = x^{3} for x ≥ 0
 2.6 Tangent Lines and Their Slopes

 Define slope of tangent line as limit of slopes of secant lines
 Find slope of tangent line to y = x^{2} at (1,1)
 Find slope of tangent line to y = √x at x = 4
 Find the equation for the tangent line of 4x^{2} + 10x + 5 at the point (2,1) using the limit definition of the derivative
 2.7 The Derivative

 Compute the derivative of f(x) = √(x + 2) using the limit definition of derivative
 2.8 Differentiation Rules

 Derivative of x^{n} is nx^{n–1}
 Find derivative of t^{17} at t = 1
 find f'(8) if f(x) = cuberoot of x
 Statement of the product rule
 Find derivative of (3x + 2)x^{1/2}
 Find derivative of sin(x)(x^{3} + 1)
 Statement of the quotient rule
 Find derivative of (x + 3)/(x^{2} + 7)
 Find derivative of tan(x)
 Statement of the chain rule, viewing the rule as describing related rates of change, example: rate of change of volume of balloon when radius changes. This video is 18+ minutes
 Describing the velocity of a duck flying along a parameterized curve. This video is 17+ minutes
 Statement of the chain rule, examples of compositions of functions
 Find derivative of (x^{3} + 1)^{1/2}
 2.9 Derivatives of the Trigonometric Functions

 Find derivative of cos(x^{2})
 Compute the derivative of x^{2}·sin(√x)
 2.10 The Mean Value Theorem

 Tangent is positive where function is increasing, negative where function is decreasing
 3x^{2} + 12x
 x^{3} + 17 – 12x
 2.11 Implicit Differentiation

 The second derivative is the derivative of the derivative
 sin(x^{2} + 2)
 sin(2x)
 2.12 Derivatives of Exponential and Logarithm Functions

 Use chain rule and d/dx (e^{x}) = e^{x} to solve problems
 Find derivative of y = x^{2}·e^{3x}
 Why can't I use power rule on e^{x}?
 Find derivative of y = ln(x^{7} + 3x + 2) if x>0
 Compute the derivative of e^{x^2}/tan(x)
 2.13 Newton's Method

 Newtons method
 Newton's method might go wrong
 Newton's method bites the dust
 2.14 Linear Approximations

 Find the equation for the tangent line of 4x^{2} + 10x + 5 at the point (2,1) using power rule to find formula for derivative
 cos(x) at π/3
 2.15 Antiderivatives and Initial Value Problems

 If f(x) is an antiderivative of f'(x), so is f(x) + constant
 y'(x) = cos(3x), y(0) = 5
 y'(x) = x^{12} + 1, y(1) = 15/13
 2.16 Velocity and Acceleration

 Ball is thrown upward 10 m/s initial velocity; how high does it go?
 Deriving s(t) = 16t^{2} + v_{0}t + s_{0}
 How long does it take Wile E. Coyote to fall off a 100ft cliff?
 If Wile E. Coyote takes 20 seconds to reach the ground, how high is the cliff?
 If initial velocity is 3 ft/s upward, and he hits the ground with velocity 100 ft/s, how high is the cliff?
 2.17 Related Rates

 2.18 Case Study: Torricelli's Law

3 Modeling with Differential Equations
 3.1 Introduction to the Issues

 What is a differential equation?
 3.2 Exponential Growth and Decay


Radioactive decay in Earth Science determines approximate age of rocks
 3.3 Separable Differential Equations

 Separation of variables dP/dt = kP
 3.4 Slope Fields and Euler's Method

 3.5 Issues in Curve Sketching

 Determine properties of a function from its graph
 Sketch graph of (x – 1)/(x + 1)
 Sketch graph of (x^{2} – 1)/(x^{2} + 1)
 Sketch the derivative (1)
 Sketch the derivative (2)
 3.6 Optimization

 3.7 Case Study: Population Modeling

4 Modeling Accumulations
 4.1 Introduction to the Issues

 4.2 The Definite Integral

 3 + 5 + 7 + 9 + 11 = summation from i = 0 to 4 of (3 + 2i)
 – 1/2 + 2/4 – 3/8 + 4/16 – 5/32 + ... + – 17/(2^{17})
 Approximate area under a curve by adding areas of rectangles
 What integral equals the limit as n approaches ∞ of the summation from i = 0 to n of (1 + (2i/n)^{2}) · 2/n ?
 Estimate area under f(x) = x^{2} from x = 0 to x = 2, using 5 rectangles
 4.3 Properties of the Definite Integral

 What is the value of the integral from –3 to 0 of √(9 – x^{2})?
 4.4 The Fundamental Theorem of Calculus


Use the fundamental theorem of calculus for definite integrals
 Find the area under y = x^{4} between x = 1 and x = 5
 Find the integral from –1 to 1 of x^{3}
 4.5 Techniques of Integration

 Doing the chain rule, backwards
 Integrate e^{x}/(1 + e^{2x})
 Integrate (sin(x))^{4} ·(cos(x))^{3}
 Integrate (x + 2)/√(x^{2} + 4x + π)
 Integrate x^{27} + 3sin(x)
 Integrate Ax^{2} + Bx + C
 4.6 Trapezoid Rule

 4.7 Areas Between Curves

 x^{2} and √x
 4.8 Volumes of Solids of Revolution

 4.9 Arc Length

 4.10 Inverse Trigonometric Functions

 d/dx (arctan(x)) = 1/(1 + x^{2})
 Find derivative of arcsin(3x^{2})
 Find derivative of y = arctan(e^{x})
 4.11 Case Study: Flood Watch

5 Culminating Experience
 5.1 Case Study: Sleuthing Galileo

Videos copyright © 2001 by Edwin Gailits, Kim Rheinlander, Dorothy Wallace
Last updated September 25, 2002