MAT135 EXAM REVIEW NOTES
TOPICS
❏ Calculate Limits
❏ Infinite Limits
❏ Limits to ∞
❏ Limit Laws
❏ Squeeze Theorem
❏ Continuity
❏ Intermediate Value Theorem
❏ Tangent Line of Function
❏ Derivative Rules
❏ Differentiability
❏ Implicit Differentiation
❏ Derivative of Inverse Trigonometry
❏ Exponential Growth/Decay
❏ Related Rate
❏ Critical Points
❏ Mean Value Theorem
❏ First Derivative Test
❏ Second Derivative Test
❏ L’Hopital’s Rule
❏ Optimization
❏ Limits of Arc Trigs
❏ Rolle’s Theorem
, How to Calculate Limits
Limit - the y-value a function approaches when it approaches an x-value
f (x+h) − f (x)
Official def’n of a derivative : l lim h
h→ 0
One-Sided Limits: Only looking at one side in question
Right-Handed Limit: lim f(x) = L
x→ a+
Left-Handed Limit: lim f(x) = L
x→a−
FACT
The limit lim x→ a f(x) = L will exist iff both the right and left handed limits are equal
Limit Properties
1. Constant Rule lim[cf(x)] = c limf(x)
2. Sum or Difference lim[f(x) +- g(x)] = lim f(x) +- lim g(x)
3. Product rule lim[f(x)g(x)] = limf(x)limg(x)
f (x) limf (x)
4. Quotient rule lim [ g(x) ] = limg(x) if limg(x) =/ 0
5. Power rule lim[f(x)]n = [limf(x)]n
6. Special Power rule lim n √f (x) = √ n
limf (x)
7. Limit of a constant is the constant limc = c
8. lim x→ a = a
9. lim x→ a xn = an
Ex: Compute lim as x→ -2 (3x2+5x -9)
1. lim 3x2 + lim5x - lim9
2. 3limx2 + 5limx - lim9
3. 3(-2)2+5(-2) -9 = -7
When can You just Plug in values?
- Polynomials
- p(x) / q(x) if no division of zero
- cos(x) and sin(x)
- sec(x) and tan(x), where cos(x) =/ 0
- csc(x) and cot(x), where sin(x) =/ 0
n
- √x if n is odd
- ax and ex
- logbx and lnx for all x > 0
ex
Ex : Evaluate lim (- √5 x + 1+ln(x) +sin(x)cos(x))
x→ 3
-this is just a combination of the several functions above, so we can just plug in the value
TOPICS
❏ Calculate Limits
❏ Infinite Limits
❏ Limits to ∞
❏ Limit Laws
❏ Squeeze Theorem
❏ Continuity
❏ Intermediate Value Theorem
❏ Tangent Line of Function
❏ Derivative Rules
❏ Differentiability
❏ Implicit Differentiation
❏ Derivative of Inverse Trigonometry
❏ Exponential Growth/Decay
❏ Related Rate
❏ Critical Points
❏ Mean Value Theorem
❏ First Derivative Test
❏ Second Derivative Test
❏ L’Hopital’s Rule
❏ Optimization
❏ Limits of Arc Trigs
❏ Rolle’s Theorem
, How to Calculate Limits
Limit - the y-value a function approaches when it approaches an x-value
f (x+h) − f (x)
Official def’n of a derivative : l lim h
h→ 0
One-Sided Limits: Only looking at one side in question
Right-Handed Limit: lim f(x) = L
x→ a+
Left-Handed Limit: lim f(x) = L
x→a−
FACT
The limit lim x→ a f(x) = L will exist iff both the right and left handed limits are equal
Limit Properties
1. Constant Rule lim[cf(x)] = c limf(x)
2. Sum or Difference lim[f(x) +- g(x)] = lim f(x) +- lim g(x)
3. Product rule lim[f(x)g(x)] = limf(x)limg(x)
f (x) limf (x)
4. Quotient rule lim [ g(x) ] = limg(x) if limg(x) =/ 0
5. Power rule lim[f(x)]n = [limf(x)]n
6. Special Power rule lim n √f (x) = √ n
limf (x)
7. Limit of a constant is the constant limc = c
8. lim x→ a = a
9. lim x→ a xn = an
Ex: Compute lim as x→ -2 (3x2+5x -9)
1. lim 3x2 + lim5x - lim9
2. 3limx2 + 5limx - lim9
3. 3(-2)2+5(-2) -9 = -7
When can You just Plug in values?
- Polynomials
- p(x) / q(x) if no division of zero
- cos(x) and sin(x)
- sec(x) and tan(x), where cos(x) =/ 0
- csc(x) and cot(x), where sin(x) =/ 0
n
- √x if n is odd
- ax and ex
- logbx and lnx for all x > 0
ex
Ex : Evaluate lim (- √5 x + 1+ln(x) +sin(x)cos(x))
x→ 3
-this is just a combination of the several functions above, so we can just plug in the value
