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# 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·ex^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 ex 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 nxn–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)x1/2
• 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 (ex) = ex to solve problems
• Find derivative of y = x2·e3x
• Why can't I use power rule on ex?
• Find derivative of y = ln(x7 + 3x + 2) if x>0
• Compute the derivative of ex^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 ex/(1 + e2x)
• 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(ex)

4.11 Case Study: Flood Watch

## 5 Culminating Experience

5.1 Case Study: Sleuthing Galileo