Week 1 small class
Topics: Hill functions; sketching products and ratios of familiar functions
Learning Objectives:
Axn
• Sketch rational functions of the form by considering behaviour for small x and for large x.
(B + xn )
• Sketch products and ratios of familiar functions by separately considering component functions.
Problems and takeaways (Hill functions):
1. Consider the function
5x3
f (x) = , x ≥ 0.
9 + x3
What does the denominator behave like for very small x?
2. Sketch f (x) for small x.
3. If x is large, what does the denominator behave like?
4. Sketch f (x) for large x. Fill in the gaps to make a rough sketch of the whole function.
6x4
5. Sketch g(x) = , x > 0.
16 + x4
Axn A
6. Takeaway: Rational functions of the form B+xn resemble the power function B xn near the origin,
and the constant A far from the origin.
Problems and takeaways (Combinations of familiar functions):
1. Sketch the following functions.
x
(a) .
1 + x2
(b) ex cos x.
1
(c) x .
e +1
2. Takeaway: For more complicated functions, we can get a sense of the graph by considering the
function’s individual components.
Additional reading:
• First steps in graph sketching: Section 1.4 of Differential calculus for the life sciences
1
,Week 2 small class
Topics: Continuity
Learning Objectives:
• Explain informally what it means for a function to be continuous on its domain.
• Identify and classify points of discontinuity (jump, infinite, removable).
• Given a function defined with parameters, select parameter values that make the function continuous.
• Determine where a given function is continuous.
Problems and takeaways (Continuity):
1. What is lim x3 ? Why can we just “plug it in”?
x→2
2. Definition: We call a function continuous at x = a if lim f (x) = f (a). Continuous functions are
x→a
“nice” functions. Informally, where a function is continuous, it can be drawn without lifting your pencil
off the page.
3. Which of the following functions are continuous on their domains?
(a) polynomials
(b) sin(x), cos(x)
(c) tan(x)
(d) ex , log(x)
4. Takeaway: All of the “familiar functions” we just listed are continuous on their domains.
5. Draw a function whose domain is all real numbers that is not continuous.
See CLP-1 example 1.6.4.
6. What is a real-life example of a discontinuous function?
Problems (Making functions continuous)
x3 − 2x2
1. Sketch f (x) = .
x−2
( 3 2
x −2x
x−2 if x ̸= 2
2. Consider g(x) = . Find a such that g(x) is continuous.
a if x = 2
(
8 − kx if x < k
3. Find k such that f (x) = is continuous.
x2 if x ≥ k
4. Takeaway: If the pieces of a piecewise function are continuous, you can make the piecewise function
continuous by “matching endpoints”.
1
, Problems and takeaways (Where are functions continuous?):
1. Design a function with exactly two discontinuities. It can contain familiar functions, and it can be a
piecewise function. Your task will be to find the discontinuities in another team’s function.
2. Trade functions with another team, and find their discontinuities!
Additional problems:
• CLP-1 Problem book section 1.6: Q1, Q7-Q11, Q13-Q18, Q20.
2
Topics: Hill functions; sketching products and ratios of familiar functions
Learning Objectives:
Axn
• Sketch rational functions of the form by considering behaviour for small x and for large x.
(B + xn )
• Sketch products and ratios of familiar functions by separately considering component functions.
Problems and takeaways (Hill functions):
1. Consider the function
5x3
f (x) = , x ≥ 0.
9 + x3
What does the denominator behave like for very small x?
2. Sketch f (x) for small x.
3. If x is large, what does the denominator behave like?
4. Sketch f (x) for large x. Fill in the gaps to make a rough sketch of the whole function.
6x4
5. Sketch g(x) = , x > 0.
16 + x4
Axn A
6. Takeaway: Rational functions of the form B+xn resemble the power function B xn near the origin,
and the constant A far from the origin.
Problems and takeaways (Combinations of familiar functions):
1. Sketch the following functions.
x
(a) .
1 + x2
(b) ex cos x.
1
(c) x .
e +1
2. Takeaway: For more complicated functions, we can get a sense of the graph by considering the
function’s individual components.
Additional reading:
• First steps in graph sketching: Section 1.4 of Differential calculus for the life sciences
1
,Week 2 small class
Topics: Continuity
Learning Objectives:
• Explain informally what it means for a function to be continuous on its domain.
• Identify and classify points of discontinuity (jump, infinite, removable).
• Given a function defined with parameters, select parameter values that make the function continuous.
• Determine where a given function is continuous.
Problems and takeaways (Continuity):
1. What is lim x3 ? Why can we just “plug it in”?
x→2
2. Definition: We call a function continuous at x = a if lim f (x) = f (a). Continuous functions are
x→a
“nice” functions. Informally, where a function is continuous, it can be drawn without lifting your pencil
off the page.
3. Which of the following functions are continuous on their domains?
(a) polynomials
(b) sin(x), cos(x)
(c) tan(x)
(d) ex , log(x)
4. Takeaway: All of the “familiar functions” we just listed are continuous on their domains.
5. Draw a function whose domain is all real numbers that is not continuous.
See CLP-1 example 1.6.4.
6. What is a real-life example of a discontinuous function?
Problems (Making functions continuous)
x3 − 2x2
1. Sketch f (x) = .
x−2
( 3 2
x −2x
x−2 if x ̸= 2
2. Consider g(x) = . Find a such that g(x) is continuous.
a if x = 2
(
8 − kx if x < k
3. Find k such that f (x) = is continuous.
x2 if x ≥ k
4. Takeaway: If the pieces of a piecewise function are continuous, you can make the piecewise function
continuous by “matching endpoints”.
1
, Problems and takeaways (Where are functions continuous?):
1. Design a function with exactly two discontinuities. It can contain familiar functions, and it can be a
piecewise function. Your task will be to find the discontinuities in another team’s function.
2. Trade functions with another team, and find their discontinuities!
Additional problems:
• CLP-1 Problem book section 1.6: Q1, Q7-Q11, Q13-Q18, Q20.
2
