Principles of Calculus Modeling
An Interactive Approach

1 Modeling Discrete Data

1.1 Introduction to the Issues

• What is the sum of the squared errors?
• Find the parabola passing through (1, 1) (5, 1) and (3, 3).

1.2 Lines in the Plane

• Find the line parallel to y = 1/2 x – 1.
• Find the line perpendicular to the line connecting (6, –2) and (4, –4) that passes through the midpoint of the segment joining these points.

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

• Sum of Squared Errors

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
• Given the graph of a smooth curve, draw its derivative
• Given the graph of a curve with corners, draw its 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

• A balloon is being inflated at 1 ft³/min. At what rate is the radius increasing when the radius = 1 ft? 4 ft?
• A 6 foot man walks away from an 18 foot lamp post. At what rate does his shadow lengthen if he walks 10 feet per minute?

2.18 Case Study: Torricelli's Law

• Boyle's Law PV = kT
• Relate dP/dt to dV/dt in Boyle's Law

3 Modeling with Differential Equations

3.1 Introduction to the Issues

• What is a differential equation?
• The differential equation: y′ = 2x + 2

3.2 Exponential Growth and Decay

• Radioactive decay in Earth Science determines approximate age of rocks
• Solving the basic population growth differential equation
• A Population of bacteria doubles every 2.5 hours. What equation governs its growth?
• "The Chafing Dish"

3.3 Separable Differential Equations

• Separation of variables dP/dt = kP
• Solve the differential equation: dy/dx = (2x² + 3x)/(y + 1)

3.4 Slope Fields and Euler's Method

• Use Euler's Method to solve y′ = xy

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

Find the open box of largest volume that can be built from a 24" x 20" rectangle by removing squares from the corners.

• Sketch the rectangle to understand the problem
• Experiment with different size boxes
• The solution using calculus

Farmer Maria wishes to enclose the maximum area of pasture in a rectangle. One side of the pasture is already fenced. She has 100 feet of fence left. What dimensions should she make her field?

• Visualize the problem with different rectangles
• The solution using calculus

3.7 Case Study: Population Modeling

• Solve dw/dt = k(1 - w/m) to model population of weeds in a garden
  

4 Modeling Accumulations

4.1 Introduction to the Issues

• Write 1/5 + 3/25 + 5/125 + 7/625 + ... in sigma notation

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)?
• Simplify the integral from –a to a of sin(2x) + x.

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

• Estimate the integral of x² dx from 0 to 6 using the Trapezoid Rule with 6 trapezoids.

4.7 Areas Between Curves

• Find the area between x² and √x
• Find the area between y = x² and y = 2 – x²
• Find the area between the ellipse x²/9 + y² = 1 and the circle x² + y² = 1

4.8 Volumes of Solids of Revolution

• Find the volume of the cone created by rotating y = 3x around the x-axis between x = 0 and x = 1
• Find the volume of the solid created by rotating y = cos x around the x-axis between x = 0 and x = π/2

4.9 Arc Length

• Find the length of the curve y = cosh x from x = –1 to x = 1

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

• Find the x coordinate of the centroid of the region bounded by y = 0, y = x², x = 0, x = 1
• Find the y coordinate of the centroid of the region bounded by y = 0, y = x², x = 0, x = 1

5 Culminating Experience

5.1 Case Study: Sleuthing Galileo

Last updated August 13, 2003