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Welcome to Math 102 - Calculus for Life the Sciences
• Instructor: Prof. Eric Cytrynbaum
• Email: cytryn@math.ubc.ca
• Course website: wiki.math.ubc.ca
• Today:
• Information about the course.
• Shapes of cells and power functions.
Math 102 - info
• Homework:
• WeBWorK (online) - 10%
• Old-School Homework aka OSH (written) - 10%
Math 102 - info
• Homework:
• WeBWorK (online) - 10%
• Old-School Homework aka OSH (written) - 10%
• Quizzes - 15%
Math 102 - info
• Homework:
• WeBWorK (online) - 10%
• Old-School Homework aka OSH (written) - 10%
• Quizzes - 15%
• Midterm (Oct 26 - evening, by sign-up) - 15%
Math 102 - info
• Homework:
• WeBWorK (online) - 10%
• Old-School Homework aka OSH (written) - 10%
• Quizzes - 15%
• Midterm (Oct 26 - evening, by sign-up) - 15%
• Final exam - 50% (“44% rule”)
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
• Getting help - Piazza, Math Learning Centre, office hrs.
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
• Getting help - Piazza, Math Learning Centre, office hrs.
• Online course notes - by Leah Keshet (free pdf).
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
• Getting help - Piazza, Math Learning Centre, office hrs.
• Online course notes - by Leah Keshet (free pdf).
• Pre-lecture videos.
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
• Getting help - Piazza, Math Learning Centre, office hrs.
• Online course notes - by Leah Keshet (free pdf).
• Pre-lecture videos.
• Read over website, do “Course logistics” assignment.
Math 102 - info• WeBWorK -
• Pre-lecture (MW 7am), post-lecture (Th 7am).
• 5%-drop rule (by points).
• Getting help - Piazza, Math Learning Centre, office hrs.
• Online course notes - by Leah Keshet (free pdf).
• Pre-lecture videos.
• Read over website, do “Course logistics” assignment.
• Subscribe to due-date calendar.
Math 102 - info
• In class, you will be expected to actively think and work!
Math 102 - info
• In class, you will be expected to actively think and work!
• I will use a short-lecture format with lots of clicker questions.
Math 102 - info
• In class, you will be expected to actively think and work!
• I will use a short-lecture format with lots of clicker questions.
• Register your iClicker on Connect (once).
White blood cells (spheres)
Shapes of cells
Shapes of cells
Shapes of cells
• Cellular metabolism - cells use energy/nutrients proportional to volume but absorb them proportional to surface area.
Shapes of cells
• Cellular metabolism - cells use energy/nutrients proportional to volume but absorb them proportional to surface area.
• Need absorption rate > consumption rate to survive.
Shapes of cells
• Cellular metabolism - cells use energy/nutrients proportional to volume but absorb them proportional to surface area.
• Need absorption rate > consumption rate to survive.
• For different shapes, this balance scales better or worse as size increases...
Nutrient balance in a spherical cell
Nutrient balance in a spherical cell
• Absorption is proportional to surface area:
Nutrient balance in a spherical cell
• Absorption is proportional to surface area:
S = 4�r2 A = k1S = 4k1�r2
Nutrient balance in a spherical cell
• Absorption is proportional to surface area:
• Consumption is proportional to volume:
S = 4�r2 A = k1S = 4k1�r2
Nutrient balance in a spherical cell
• Absorption is proportional to surface area:
• Consumption is proportional to volume:
V =43�r3
S = 4�r2 A = k1S = 4k1�r2
C = k2V =43k2�r3
k1 k2where and are positive constants.
Which of the following is true?
A. Absorption is greater than consumption for sufficiently large cells and vice versa for small cells.
B. Consumption is greater than absorption for sufficiently large cells and vice versa for small cells.
C. Both A and B are possible - it depends on and .
C =43k2�r3 A = 4k1�r2
k1
k2
Which of the following is true?
A. Absorption is greater than consumption for sufficiently large cells and vice versa for small cells.
B. Consumption is greater than absorption for sufficiently large cells and vice versa for small cells.
C. Both A and B are possible - it depends on and .
C =43k2�r3 A = 4k1�r2
k1
k2
Power functions
Power functions
• A function of the form (where a is a constant and n is an integer) is called a power function.
f(x) = ax
n
Power functions
• A function of the form (where a is a constant and n is an integer) is called a power function.
• Both and are power functions (with r as independent variable).
C =43k2�r3 A = 4k1�r2
f(x) = ax
n
Power functions
• A function of the form (where a is a constant and n is an integer) is called a power function.
• Both and are power functions (with r as independent variable).
C =43k2�r3 A = 4k1�r2
f(x) = ax
n
C(r) =✓
43k2�
◆r3
Power functions
• A function of the form (where a is a constant and n is an integer) is called a power function.
• Both and are power functions (with r as independent variable).
C =43k2�r3 A = 4k1�r2
f(x) = ax
n
C(r) =✓
43k2�
◆r3 A(r) = (4k1�) r2
Power functions(A) Green: x3, yellow: x4,
red: x5, blue: x6.
(B) Green: x5, yellow: x4, red: x3, blue: x2.
(C) Green: x6, yellow: x5, red: x4, blue: x3.
(D) Either (B) or (C) (not enough info).
(E) Don’t know - please explain.
Power functions(A) Green: x3, yellow: x4,
red: x5, blue: x6.
(B) Green: x5, yellow: x4, red: x3, blue: x2.
(C) Green: x6, yellow: x5, red: x4, blue: x3.
(D) Either (B) or (C) (not enough info).
(E) Don’t know - please explain.
Limit on cell size
Limit on cell size
• When is absorption > consumption?
Limit on cell size
• When is absorption > consumption?
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?stretch r2 vertically
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?stretch r2 vertically
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?stretch r2 vertically stretch r3 vertically
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?stretch r2 vertically stretch r3 vertically
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?
• Solve for r in terms of other parameters:
stretch r2 vertically stretch r3 vertically
A = 4k1�r2 >43k2�r3 = C
Limit on cell size
• When is absorption > consumption?
• Solve for r in terms of other parameters:
stretch r2 vertically stretch r3 vertically
A = 4k1�r2 >43k2�r3 = C
r < 3k1
k2.
Limit on cell size
• When is absorption > consumption?
• Solve for r in terms of other parameters:
stretch r2 vertically stretch r3 vertically
A = 4k1�r2 >43k2�r3 = C
r < 3k1
k2.
• No matter what k values, inequality doesn’t flip.
Neuron (1 meter)
The “biggest” cells around
Caulerpa prolifera (single cell, 1 meter)
The “biggest” cells around
Getting around S:V issues
Getting around S:V issues
• How do cells get around the problem?
• Don’t be 3D, be a filament or sheet.
Kiwi egg (not the biggest but remarkable)
Exceptions
Ostrich egg
Exceptions
Extra - How does this cell get around the S:V issue?
Sailors eyeball
“Exceptions”By Haplochromis - selbst fotografiert von Haplochromis, Public
Domain, https://commons.wikimedia.org/w/index.php?curid=522246
Getting around S:V issues
• If you want to be big,
Getting around S:V issues
• If you want to be big,
• don’t be 3D, be a filament or sheets,
Getting around S:V issues
• If you want to be big,
• don’t be 3D, be a filament or sheets,
• don’t consume proportional to volume,
Getting around S:V issues
• If you want to be big,
• don’t be 3D, be a filament or sheets,
• don’t consume proportional to volume,
• make sure your k1>>k2, that is have highly permeable shell,
Getting around S:V issues
• If you want to be big,
• don’t be 3D, be a filament or sheets,
• don’t consume proportional to volume,
• make sure your k1>>k2, that is have highly permeable shell,
• store everything you need inside (eggs).
Getting around S:V issues
• If you want to be big,
• don’t be 3D, be a filament or sheets,
• don’t consume proportional to volume,
• make sure your k1>>k2, that is have highly permeable shell,
• store everything you need inside (eggs).
• Predictions of the model are only as good as the validity of our assumptions.
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