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MATHEMATICS โ Differential Calculus 1
Vernante 9
LIMIT OF A FUNCTION
Limit is a number such that the value of a given
function remains arbitrarily close to this number
when the independent variable is sufficiently close to
a specified point.
Theorems on Limits
THE DERIVATIVE
The derivative of ๐ฆ is the limit of the ratio of the
incremental change of ๐ฆ to the incremental change of
๐ฅ as the incremental change of ๐ฅ approaches zero. In
symbol:
โฒ
= lim
โ๐ฅโ 0
= lim
โโ 0
The following are some other ways of writing the
derivative of ๐ฆ:
โฒ
โฒ
Differentiation of Algebraic Functions
Differentiation of Trigonometric Functions
Differentiation of Inverse Trigonometric
Functions
MATHEMATICS โ Differential Calculus 1
Vernante 9
Differentiation of Logarithmic Functions
Differentiation of Exponential Functions
SAMPLE PROBLEMS
- What is the value of lim
๐ฅโ 2
2
a. - 3 c. - 1
b. 3 d. 1
- Evaluate: lim
๐ฅโ 4
๐ฅ
3
โ 64
๐ฅ
2
โ 16
a. - 6 c. 0
b. 6 d. undefined
- Evaluate: lim
๐ฅโ 0
1 โcos ๐ฅ
๐ฅ
2
a. 0 c. - 1/
b. 2 d. 1/
- Find ๐๐ฆ/๐๐ฅ if ๐ฆ = 3 ๐ฅ
3
2
๐ฅ
a. 9 ๐ฅ
2
2
๐ฅ
๐ฅ
c. 9 ๐ฅ
2
2
๐ฅ
b. 9 ๐ฅ
2
2
๐ฅ
๐ฅ
d. 9 ๐ฅ
2
2
๐ฅ
- Find the derivative of ๐ฆ =
4 ๐ฅโ 5
2 ๐ฅ+ 1
a. ๐ฆ
โฒ
18
( 2 ๐ฅ+ 1 )
2
c. ๐ฆ
โฒ
14
( 2 ๐ฅ+ 1 )
2
b. ๐ฆ
โฒ
โ 14
( 2 ๐ฅ+ 1 )
2
d. ๐ฆ
โฒ
โ 18
( 2 ๐ฅ+ 1 )
2
- Find ๐๐ฆ/๐๐ก if ๐ฆ = ๐ฅ
2
- 3 ๐ฅ + 1 and ๐ฅ = ๐ก
2
a. 4 ๐ก
3
3
b. 4 ๐ก
3
3
2
- Find the derivative of ๐ฆ = โ
a.
1
โ
5 โ 6 ๐ฅ
c.
3
โ
5 โ 6 ๐ฅ
b.
โ 2
โ
5 โ 6 ๐ฅ
d.
โ 3
โ
5 โ 6 ๐ฅ
- Find the derivative of ๐ฆ = sin 5 ๐ฅ โ
1
3
sin
3
a. ๐ฆ
โฒ
= sin
3
5 ๐ฅ c. ๐ฆ
โฒ
= 5 sin
3
b. ๐ฆ
โฒ
= 5 cos
3
5 ๐ฅ d. ๐ฆ
โฒ
= cos
2
- What is the first derivative of ๐ฆ = arc sin 3 ๐ฅ?
a.
โ 3
1 + 9 ๐ฅ
2
c.
โ 3
โ 1 โ 9 ๐ฅ
2
b.
3
1 + 9 ๐ฅ
2
d.
3
โ 1 โ 9 ๐ฅ
2
- Find the point in the parabola ๐ฆ
2
= 4 ๐ฅ at which
the rate of change of the abscissa and ordinate are
equal.
a. (1,2) c. (4,4)
b. (2,1) d. (-1,4)
- Find the equation of the line tangent to the curve
๐ฅ at (4,7).
a. 3 ๐ฅ โ 4 ๐ฆ + 16 = 0 c. 3 ๐ฅ โ 4 ๐ฆ โ 16 = 0
b. 3 ๐ฅ + 4 ๐ฆ + 16 = 0 d. 3 ๐ฅ + 4 ๐ฆ โ 16 = 0
- Find the equation of the normal to the curve
2
โ 2 ๐ฅ + 2 at (2,10).
a. ๐ฅ + 5 ๐ฆ โ 52 = 0 c. ๐ฅ + 10 ๐ฆ โ 102 = 0
b. ๐ฅ โ 10 ๐ฆ + 98 = 0 d. ๐ฅ โ 5 ๐ฆ + 48 = 0
- Find the slope of the tangent to the curve
2
2
- 4 through the point (-2,4).
a. - 3/2 c. 2/
b. - 2/3 d. 3 / 2
- Evaluate the first derivative of the implicit
function: 4 ๐ฅ
2
2
a.
4 ๐ฅ+๐ฆ
๐ฅ+๐ฆ
c.
4 ๐ฅโ๐ฆ
๐ฅ+๐ฆ
b. โ
4 ๐ฅ+๐ฆ
๐ฅ+๐ฆ
d. โ
4 ๐ฅ+๐ฆ
๐ฅโ๐ฆ
- Find the slope of the ellipse ๐ฅ
2
2
16 ๐ฆ + 5 = 0 at the point where ๐ฆ = โ 2 + 8
- 5
and
a. - 0.1654 c. - 0.
b. - 0.1538 d. - 0. 1768
- If ๐ฆ = ๐ฅ
ln ๐ฅ
, find
๐
2
๐ฆ
๐๐ฅ
2
a. 1 /๐ฅ
2
c. 1 /๐ฅ
b. - 1 /๐ฅ d. - 1 /๐ฅ
2
- How fast does the slope of the curve
2
2
change when ๐ฅ = 1?
a. 30 c. 35
b. 25 d. 40
Situation 1. For problems 18-20, refer here. Given the
function โ
3
2
4
6
5
, find:
๐โ
๐๐ข
a. 30 ๐ข
4
5
c. - 10 ๐ฃ๐ฆ
4
b. ๐ฆ
5
4
d. 24 ๐ข
2
2
3
6
๐โ
๐๐ฅ
a. 30 ๐ข
4
5
c. - 10 ๐ฃ๐ฆ
4
b. ๐ฆ
5
4
d. 24 ๐ข
2
2
3
6
๐โ
๐๐ฆ
a. 30 ๐ข
4
5
c. - 10 ๐ฃ๐ฆ
4
b. ๐ฆ
5
4
d. 24 ๐ข
2
2
3
6
