This document provides a review package for a Math 196 final exam, including formulas, trigonometry rules, calculus techniques like integration by substitution and using the Fundamental Theory of Calculus, how to sketch functions, solve rate of change problems, calculate surface areas, and optimize functions. It is produced by the student group UToronto SOS (Students Offering Support) to help raise student marks.

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MATH 196
FINAL EXAM-AID
REVIEW PACKAGE
Tutors: Arman Ghaffari, arman.ghaffari@mail.utoronto.ca
Roya Rahnejat roya.r1994@gmail.com
utsg.soscampus.com

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Formulas
Surface
Circle: 𝜋𝑟!
Sphere: 4𝜋𝑟!
Cylinder: 2 𝜋𝑟ℎ
Volume:
Cylinder: 𝜋𝑟!
ℎ
Cone:
!!!!
!
Sphere:
!!!!
!
Trigonometry
𝑆𝑖𝑛!
𝑥 + 𝑐𝑜𝑠!
𝑥 = 1
sin 2𝑥 = 2sin (𝑥)𝑐𝑜𝑠(𝑥)
cos (2𝑥)=1-2𝑠𝑖𝑛!
𝑥 = 2𝑐𝑜𝑠!
𝑥 − 1 = 𝑐𝑜𝑠!
(𝑥)−𝑠𝑖𝑛!
(𝑥)
𝑡𝑎𝑛!
𝑥 + 1 = 𝑠𝑒𝑐!
𝑥 =
1
𝑐𝑜𝑠!(𝑥)

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Integration by Substitution
“U” substitution
When to use:
1. Usually for “ex
”, take the whole equation as “u”.
2. For trigonometric equations such as sinx and cosx. Usually take
the higher power as “u”.
Hints:
• Having cosx and sinx inside one integral.
• Having ex
+c and alone ex
in an equation.
• High degrees!
How:
Search for parameter which is a derivative of another one!
1. Compute the following
(reminder ln6 is just a number!)
𝑒!!
1 + 𝑒! 𝑑𝑥
!"!
!

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2. Compute the following
sin!
𝑥 𝑑𝑥
reminder
sin!
𝑥 + cos!
𝑥 = 1
𝑑(𝑠𝑖𝑛𝑥)
𝑑𝑥
= 𝑐𝑜𝑠𝑥
Fundamental Theory of Calculus
When do we use Fundamental Theory of Calculus?
1. Taking derivative of an integral
2. Having variables in the boundary of your integral
d
dx
𝑓 𝑡 𝑑𝑡
! !
! !
= 𝑓 ℎ 𝑥 ℎ!
𝑥 − 𝑓 𝑔 𝑥 𝑔′(𝑥)

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3. Compute
lim
!→!
5𝑥!
+ 3𝑥!
3𝑡 + cos 𝑡! 𝑑𝑡
!!
!

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Volume Integrals
When to use each:
4. a) Find the volume obtained by revolving the curve y=x1/2
around x
axis for [0,1]

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b) Find the volume obtained by revolving the curve y=x1/2
around y=-1 for
[0,1]
c) Find the volume obtained by revolving the region boundary of y=x1/2
and y=x for [0,1] about the y-axis
d) Find the volume obtained by revolving the region boundary of y=x1/2
and y=x for [0,1] about x=-1

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Rolle’s Theorem and the Mean Value Theorem
If f is continues on [a,b] and differentiable on (a,b) and f(a)=f(b) then
there is a number c so that f’(c)=0
f!
c =
𝑓 𝑏 − 𝑓 𝑎
𝑏 − 𝑎
Mean Value Theorem Consequences:
1. If f’(x)=0 for all x in the interval (a,b), then f is constant over the
interval.
2. If f’(x)>0 for all x in the interval (a,b) then f is increasing over the
interval.
3. If f’(x)<0 for all x in the interval (a,b) then f is decreasing over the
interval.
Problem 8 (2012 Final exam)
Let f be continuous on [0,1].
Suppose 𝑓 𝑥 𝑑𝑥 = 0
!
!
and f’(x)>0 on (0,1).
a) Show that f(x)=0 has a solution in [0,1]. That is, there exists
𝑥! ∈ 0,1 such that f(𝑥!) = 0.
b) Show that 𝑥! is the only solution of f(x)=0 in [0,1].

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Sketching functions
Steps:
• Find Domain
• Find Intercepts
• Horizontal Asymptote:
– Limit of f(x) as x -> ∞
• Critical Points: Where limit of f(x) is zero or does not exist
• Concavity: Inflection Points, use f”(x), the curve is concave up
where f’’(c) >0 and concave down where f’’(c) < 0
• Second derivative test: local min at c: f’(c) = o and f’’(c) >0, local
max where f’(c) = 0 and f”(c) <0
5. Sketch graph of f(x) = xe-x2

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6. Sketch a graph of
!!!
(!!!!)
Rate of Change Problems
• Draw a diagram if possible
• Express the given information and required rate in terms of
derivatives
• Write an equation that relates the quantities of the problem, use
geometry to eliminate one variable by substitution
• Use Chain rule dy/dx=dy/du .du/dx
7. If water enters a hemispherical bowl of radius 50 cm at a rate of 10
cm3
/s, how fast will the water level be rising when the depth of water in
the bowl is 20 cm?

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Surface Area
• Find the interval
• If integrating by x, find the function in terms of x
• If integrating by y, find the function in terms of y
8. Let A be the area of the region in xy plane bound by y = lnx , x = 1,
and y= 2. Find the area of A in two ways: one with respect to x, one with
respect to y.

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Optimization Problems
• Find Domain
• Find Intercepts
• Horizontal and vertical Asymptote:
– For horizontal asymptote find Limit of f(x) as x -> ∞
– The line x = a is called a vertical asymptote if lim f(x) -> ∞
As we approach a from left or right.
• Critical Points: Where limit of f(x) is zero or does not exist
• Concavity: Inflection Points, use f”(x), the curve is concave up
where f’’(c) >0 and concave down where f’’(c) < 0
• Second derivative test: local min at c: f’(c) = o and f’’(c) >0, local
max where f’(c) = 0 and f”(c) <0
Example
Find the area of the largest rectangle that can be inscribed in a
semicircle of radius r.