Principles of Calculus Modeling
An Interactive Approach

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/(x³ + x⁵)
• x·e^{x^2}
• cos(2x) + 3
• sin(x) + x²

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(x³) – 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(3x²) = 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 x²

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 right-handed 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/(x² – 16)
• limit as x approaches 7 from the right of (3x)/(x² – 8x + 7)
• limit as x approaches 7 from the left of (x – 7)/(x² – 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) = x³ 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 = x² at (1,1)
• Find slope of tangent line to y = √x at x = 4
• Find the equation for the tangent line of 4x² + 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 xⁿ is nx^n–1
• Find derivative of t¹⁷ 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)(x³ + 1)
• Statement of the quotient rule
• Find derivative of (x + 3)/(x² + 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 (x³ + 1)^1/2

2.9 Derivatives of the Trigonometric Functions

• Find derivative of cos(x²)
• Compute the derivative of x²·sin(√x)

2.10 The Mean Value Theorem

• Tangent is positive where function is increasing, negative where function is decreasing
• 3x² + 12x
• x³ + 17 – 12x

2.11 Implicit Differentiation

• The second derivative is the derivative of the derivative
• sin(x² + 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 = x²·e^3x
• Why can't I use power rule on e^x?
• Find derivative of y = ln(x⁷ + 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 4x² + 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) = x¹² + 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) = 16t² + v₀t + s₀
• How long does it take Wile E. Coyote to fall off a 100-ft 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 (x² – 1)/(x² + 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/(2¹⁷)
• 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) = x² 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 – x²)?

4.4 The Fundamental Theorem of Calculus

• Use the fundamental theorem of calculus for definite integrals
• Find the area under y = x⁴ between x = 1 and x = 5
• Find the integral from –1 to 1 of x³

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)/√(x² + 4x + π)
• Integrate x²⁷ + 3sin(x)
• Integrate Ax² + Bx + C

4.6 Trapezoid Rule

4.7 Areas Between Curves

• x² and √x

4.8 Volumes of Solids of Revolution

4.9 Arc Length

4.10 Inverse Trigonometric Functions

• d/dx (arctan(x)) = 1/(1 + x²)
• Find derivative of arcsin(3x²)
• Find derivative of y = arctan(e^x)

4.11 Case Study: Flood Watch

5 Culminating Experience

5.1 Case Study: Sleuthing Galileo

Last updated September 25, 2002