Week 1 Large Class Learning Objectives
September 13, 14, 15 (2022)
Topics: Comparing power, log, and exponential functions; trigonometric functions; parse trees
Learning Objectives:
• Sketch basic power functions.
• Determine which term in a polynomial function will dominate for small x and for large x.
• Sketch two-term polynomial functions by determining which term dominates for small x and for large
x. For example, sketch f (x) = x2 − x4 .
√
• Sketch familiar functions such as ex , log x, sin x, cos x, tan x, 1/x, x, and |x|.
• Demonstrate that ex eventually dominates any given power function and use this fact for sketching.
For example, sketch f (x) = ex − x4 .
• Given a complicated function, describe it using a parse tree.
• Given a complicated function, describe in words the order in which to apply operations.
Textbook sections:
• First steps in graph sketching: Section 1.4 of Differential calculus for the life sciences
• Parse trees: CLP-1 Section 0.5
Additional problems:
1. Exercises 1.4, 1.5, 1.6, 1.13, 1.26 from Differential calculus for the life sciences
2. Additional problems on asymptotic reasoning and sketching.
3. The Lennard-Jones potential
12 6 !
R R
V (r) = ε −2 , ε, R > 0
r r
describes the potential energy of a diatomic molecule where the atoms are a distance r > 0 apart.
(a) Write V (r) as a rational function. A rational function is the quotient of two polynomials.
(b) What function is a good approximation for V (r) for small r? For large r?
(c) Draw a rough sketch of V (r).
4. The Morse potential
2
M (r) = ε 1 − e−a(r−R) , ε, a, R > 0
is an alternative to the Lennard-Jones potential.
(a) What function is a good approximation for M (r) for small r? For large r?
(b) Draw a rough sketch of M (r).
1
, Week 2 Large Class Learning Objectives
September 20, 21, 22 (2022)
Topics: Limits; horizontal and vertical asymptotes
Learning Objectives:
• Explain using both words and pictures what lim f (x) = L, lim− f (x) = L, and lim+ f (x) = L mean
x→a x→a x→a
(including the case where L is equal to ∞ or −∞).
• Explain using using both words and pictures what lim f (x) = L and lim f (x) = L mean (including
x→∞ x→−∞
the case where L is equal to ∞ of −∞).
• Find the limit of a function at a point given the graph of the function.
• Evaluate limits of polynomial, rational, trigonometric, exponential, and logarithmic functions.
• Explain using both informal language and the language of limits what it means for a function to have
a horizontal or vertical asymptote.
• Given a simple function, find its vertical and horizontal asymptotes by asymptotic reasoning or by
taking limits.
• Use limits to find information about the shape of a function and in some cases produce a sketch.
• Explain why it is not true that a function cannot cross its horizontal asymptote.
Textbook sections:
• The Limit of a Function: CLP-1 Section 1.3
• Limits at Infinity: CLP-1 Section 1.5
• Domain, Intercepts and Asymptotes: CLP-1 Section 3.6.1
Additional Problems:
1. CLP-1 Problem Book Section 1.3: Q1-Q17.
2. CLP-1 Problem Book Section 1.5: Q1-Q7, Q27.
3. CLP-1 Problem Book Section 1.5: Q8, Q13-Q15, Q17-Q19 — but solve these problems using asymptotic
reasoning (investigating the behaviour of functions by determining which terms are “dominant”).
4. CLP-1 Problem Book Section 3.6.1: Q1, Q4, Q5.
1
September 13, 14, 15 (2022)
Topics: Comparing power, log, and exponential functions; trigonometric functions; parse trees
Learning Objectives:
• Sketch basic power functions.
• Determine which term in a polynomial function will dominate for small x and for large x.
• Sketch two-term polynomial functions by determining which term dominates for small x and for large
x. For example, sketch f (x) = x2 − x4 .
√
• Sketch familiar functions such as ex , log x, sin x, cos x, tan x, 1/x, x, and |x|.
• Demonstrate that ex eventually dominates any given power function and use this fact for sketching.
For example, sketch f (x) = ex − x4 .
• Given a complicated function, describe it using a parse tree.
• Given a complicated function, describe in words the order in which to apply operations.
Textbook sections:
• First steps in graph sketching: Section 1.4 of Differential calculus for the life sciences
• Parse trees: CLP-1 Section 0.5
Additional problems:
1. Exercises 1.4, 1.5, 1.6, 1.13, 1.26 from Differential calculus for the life sciences
2. Additional problems on asymptotic reasoning and sketching.
3. The Lennard-Jones potential
12 6 !
R R
V (r) = ε −2 , ε, R > 0
r r
describes the potential energy of a diatomic molecule where the atoms are a distance r > 0 apart.
(a) Write V (r) as a rational function. A rational function is the quotient of two polynomials.
(b) What function is a good approximation for V (r) for small r? For large r?
(c) Draw a rough sketch of V (r).
4. The Morse potential
2
M (r) = ε 1 − e−a(r−R) , ε, a, R > 0
is an alternative to the Lennard-Jones potential.
(a) What function is a good approximation for M (r) for small r? For large r?
(b) Draw a rough sketch of M (r).
1
, Week 2 Large Class Learning Objectives
September 20, 21, 22 (2022)
Topics: Limits; horizontal and vertical asymptotes
Learning Objectives:
• Explain using both words and pictures what lim f (x) = L, lim− f (x) = L, and lim+ f (x) = L mean
x→a x→a x→a
(including the case where L is equal to ∞ or −∞).
• Explain using using both words and pictures what lim f (x) = L and lim f (x) = L mean (including
x→∞ x→−∞
the case where L is equal to ∞ of −∞).
• Find the limit of a function at a point given the graph of the function.
• Evaluate limits of polynomial, rational, trigonometric, exponential, and logarithmic functions.
• Explain using both informal language and the language of limits what it means for a function to have
a horizontal or vertical asymptote.
• Given a simple function, find its vertical and horizontal asymptotes by asymptotic reasoning or by
taking limits.
• Use limits to find information about the shape of a function and in some cases produce a sketch.
• Explain why it is not true that a function cannot cross its horizontal asymptote.
Textbook sections:
• The Limit of a Function: CLP-1 Section 1.3
• Limits at Infinity: CLP-1 Section 1.5
• Domain, Intercepts and Asymptotes: CLP-1 Section 3.6.1
Additional Problems:
1. CLP-1 Problem Book Section 1.3: Q1-Q17.
2. CLP-1 Problem Book Section 1.5: Q1-Q7, Q27.
3. CLP-1 Problem Book Section 1.5: Q8, Q13-Q15, Q17-Q19 — but solve these problems using asymptotic
reasoning (investigating the behaviour of functions by determining which terms are “dominant”).
4. CLP-1 Problem Book Section 3.6.1: Q1, Q4, Q5.
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