Appendix D Examples
1 Introduction to the course
Example 1.4 Implied domain
Example 1.6 Range versus codomain
2 Limits: informal definition
Example 2.2 Function behavior
Example 2.7 Limit description of graph
Example 2.8 Limit as compared to value
Example 2.16 Using limit rules
Example 2.18 Polynomial evaluation
Example 2.19 Quotient rule does not apply
3 Limits: algebraic technique and sandwich theorem
Example 3.1 Algebraic limit technique: factoring
Example 3.2 Algebraic limit technique: clear denominator
Example 3.3 Algebraic limit technique: radicals
Example 3.7 Sandwich theorem: easy
Example 3.8 Sandwich theorem: less easy
4 Limits: formal definition
Example 4.5 Epsilon-delta: affine function
Example 4.9 Epsilon-delta: radical function
5 One-sided limits
Example 5.2 Visualizing one-sided limits
Example 5.7 The sign function
Example 5.9 Classic trig limits
6 Continuity: definition
Example 6.5 Continuity at endpoint
Example 6.7 Continuity: graphical
Example 6.9 Continuous everywhere
Example 6.13 Continuity rules
7 Continuity: intermediate value theorem
Example 7.1 Continuity: piecewise
Example 7.5 Sine equation
Example 7.6 Roots of polynomials
Example 7.7 Complicated equation
Example 7.8 Range of sine
8 Limits at infinity
Example 8.6 Limit at infinity: \(\cos\)
Example 8.11 Limit at infinity: rational function
Example 8.12 Limit at infinity: radical function
9 Infinite limits
Example 9.5 Vertical asymptotes: \(\tan\)
Example 9.9 Infinite limit: elementary examples
Example 9.10 Infinite limit: polynomial
Example 9.12 Asymptotes: rational function
10 Derivative: at a point
Example 10.4 Derivative at point: elementary
Example 10.7 Tangent line at point
11 Derivative: function
Example 11.2 Derivative function: radical
Example 11.5 Derivative function: absolute value
Example 11.7 Derivative function: \(\frac{1}{x}\)
12 Derivative: rules
Example 12.3 Power function formula: radicals
Example 12.7 Derivative rules
Example 12.10 Higher-order derivatives
13 Derivative as rate of change
Example 13.2 Height versus age
Example 13.4 Drone
Example 13.6 Marginals
14 Derivative: trig functions
Example 14.2 Tangent lines: sinusoidal
15 Chain rule
Example 15.3 Chain rule
Example 15.4 Inflating a balloon
16 Implicit differentiation
Example 16.3 Implicit differentiation: circle
Example 16.5 Tschirnhausen cubic
Example 16.6 Ellipse tangents
Example 16.7 Implicit: trig equation
17 Related rates I
Example 17.1 Inflating a balloon (redux)
Example 17.2 Sliding ladder
Example 17.5 Sliding ladder: angle of inclination
Example 17.8 Stretched rubber sheet
18 Related rates II
Example 18.1 Two trains
Example 18.3 Using similar triangles
19 Linearization
Example 19.3 Linearization: quadratic
Example 19.8 Linear approximation: quadratic
Example 19.9 Linear approximation: cube-root
Example 19.10 Linear approximation: trig
Example 19.11 Linear approximation: marshmallow
20 Extreme value theorem
Example 20.2 Extreme values: discontinuous
Example 20.4 Extreme values: open interval
Example 20.9 Local and absolute extreme values
Example 20.14 Extreme values: factored polynomial
Example 20.15 Extreme values: trig
Example 20.16 Distance between graphs
21 Mean value theorem
Example 21.4 Speeding motorist
Example 21.7 Number of roots
Example 21.9 Interesting inequality
22 Monotonicity and first derivative test
Example 22.3 Monotonicity: reciprocal
Example 22.4 Monotonicity: polynomial
Example 22.8 Classifying critical points: polynomial
Example 22.9 Monotonicity and critical points: trig
Example 22.11 Monotonicity and critical points: radical
23 Concavity and inflection points
Example 23.1 Logistic growth
Example 23.8 Concavity: radical function
Example 23.9 Concavity: polynomial
Example 23.11 Second derivative test: polynomial
24 Curve sketching I
Example 24.2 Curve sketch: polynomial
Example 24.3 Curve sketching: rational function
25 Curve sketching II
Example 25.1 Curve sketch: trig
Example 25.2 Curve sketching: rational function
26 Applied optimization
Example 26.1 Box of maximum volume
Example 26.5 Minimal fence perimeter
Example 26.8 Minimal fence perimeter (reprise)
Example 26.9 Closest point on parabola
27 Applied optimization II
Example 27.1 Snowy campus walk
Example 27.3 Optimizing revenue
