1. Capstone Project
Science, Technology, Engineering, and Mathematics
Basic Calculus
Science, Technology, Engineering, and Mathematics
Lesson 4.1
Slope and Equation of a
Tangent Line
2. 2
When a ball is thrown
upward, it follows a
parabolic path.
How do we find the
average velocity of the
ball for a certain
period of time?
3. 3
How about its velocity
at a specific time?
For example, what is
its velocity at exactly 1
second?
4. 4
This is called instantaneous velocity.
To solve for instantaneous velocity, we need
to understand slopes of tangent lines.
5. 5
How do we determine the slope
of a tangent line?
6. Learning Competency
At the end of the lesson, you should be able to do the following:
6
Illustrate the tangent line to the graph of a
function at a given point (STEM_BC11D-IIIe-1).
7. Learning Objectives
At the end of the lesson, you should be able to do the following:
7
● Find the slope of the tangent line to a curve.
● Determine the equation of the tangent line.
8. 8
Tangent Line
- a line that “just touches” the curve
and has the same direction as the
graph at the point of tangency
Secant Line
- a line that cuts through the curve
Slope of the Tangent Line
9. 9
Slope of the Tangent Line
Slope of a Line
Given two points on the line,(𝑥1, 𝑦1) and 𝑥2, 𝑦2 , the slope
of the line is given by
𝑚 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
.
10. 10
Slope of the Tangent Line
Example: What is the slope of the secant line to the
function 𝑓 𝑥 = 𝑥2
which passes through the points (1, 1)
and (2, 4)?
11. 11
Slope of the Tangent Line
Determine the slope of
each secant line of
𝑓 𝑥 = 𝑥2 which passes
through (1, 1) and each
point on the table.
(𝒙, 𝒚)
Slope of secant
line
(1.5, 2.25)
(1.25, 1.5625)
(1.2, 1.44)
(1.15, 1.3225)
(1.10, 1.21)
(1.05, 1.1025)
(1.001, 1.002001)
12. 12
Slope of the Tangent Line
Notice that the given points
approach the point (1, 1).
(𝒙, 𝒚)
(1.5, 2.25)
(1.25, 1.5625)
(1.2, 1.44)
(1.15, 1.3225)
(1.10, 1.21)
(1.05, 1.1025)
(1.001, 1.002001)
13. 13
Slope of the Tangent Line
Determine the slope of
each secant line of
𝑓 𝑥 = 𝑥2 which passes
through (1, 1) and each
point on the table.
(𝒙, 𝒚)
Slope of secant
line
(1.5, 2.25) 2.5
(1.25, 1.5625) 2.25
(1.2, 1.44) 2.2
(1.15, 1.3225) 2.15
(1.10, 1.21) 2.1
(1.05, 1.1025) 2.05
(1.001, 1.002001) 2.001
14. 14
Slope of the Tangent Line
The points approach
1, 1 .
What values do the
slopes of the secant lines
approach?
(𝒙, 𝒚)
Slope of secant
line
(1.5, 2.25) 2.5
(1.25, 1.5625) 2.25
(1.2, 1.44) 2.2
(1.15, 1.3225) 2.15
(1.10, 1.21) 2.1
(1.05, 1.1025) 2.05
(1.001, 1.002001) 2.001
15. 15
Slope of the Tangent Line
As the points approach
1,1 , what value do the
slopes of the secant lines
approach?
(𝒙, 𝒚)
Slope of secant
line
(1.5, 2.25) 2.5
(1.25, 1.5625) 2.25
(1.2, 1.44) 2.2
(1.15, 1.3225) 2.15
(1.10, 1.21) 2.1
(1.05, 1.1025) 2.05
(1.001, 1.002001) 2.001
16. 16
Slope of the Tangent Line
The slope of the tangent line of
𝑓 𝑥 = 𝑥2 at the point (1, 1) is the
limit of the slopes of the secant
lines as the points approach (1, 1).
Slope of tangent line = 2
18. 18
Equation 1: Slope of the Tangent Line
The slope of the tangent line
to the graph of the function
𝑓(𝑥) at the point (𝑎, 𝑓 𝑎 ) is
given by
𝒎 = 𝐥𝐢𝐦
𝒙→𝒂
𝒇 𝒙 − 𝒇(𝒂)
𝒙 − 𝒂
provided that this limit exists.
28. 28
Slope of the Tangent Line
Equation 1: Slope of the Tangent Line
𝒎 = 𝐥𝐢𝐦
𝒙→𝒂
𝒇 𝒙 − 𝒇(𝒂)
𝒙 − 𝒂
Let (𝑥, 𝑓 𝑥 ) be a random point on the curve of 𝑓(𝑥) such
that 𝑥 is ℎ units away from 𝑎, i.e. 𝑥 = 𝑎 + ℎ.
29. 29
Slope of the Tangent Line
Thus, we have ℎ = 𝑥 − 𝑎. Consequently, as
𝑥 approaches 𝑎, ℎ approaches 0.
30. 30
Equation 2: Slope of the Tangent Line
The slope of the tangent line
to the graph of the function
𝑓(𝑥) at the point (𝑎, 𝑓 𝑎 ) is
given by
𝒎 = 𝐥𝐢𝐦
𝒉→𝟎
𝒇 𝒂 + 𝒉 − 𝒇(𝒂)
𝒉
provided that this limit exists.
39. Try It!
39
39
Determine the slope of the tangent line
to the function 𝒇 𝒙 =
𝒙+𝟑
𝒙+𝟐
at the point
(−𝟏, 𝟐) using Equation 2.
40. 40
To solve for the equation of a line we can use the
point-slope form
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
where 𝒎 is the slope and 𝒙𝟏, 𝒚𝟏 is a point on the
line.
Equation of the Tangent Line
41. 41
After this, we use the slope-intercept form to
express the equation of the line.
𝑦 = 𝑚𝑥 + 𝑏
Equation of the Tangent Line
42. 42
In the equation of a line
𝒚 = 𝒎𝒙 + 𝒃, what values are
being represented by 𝒎 and 𝒃?
43. 43
Equation of the Tangent Line
From the previous example the
tangent line to 𝑓 𝑥 = 𝑥2 at
1,1 has a slope of 2.
44. 44
Equation of the Tangent Line
Equation of the tangent line:
𝑦 − 1 = 2 𝑥 − 1
𝑦 − 1 = 2𝑥 − 2
𝑦 = 2𝑥 − 2 + 1
𝒚 = 𝟐𝒙 − 𝟏
45. Let’s Practice!
45
Determine the equation of the tangent line to
the function 𝒇 𝒙 = 𝒙𝟐
− 𝟒 at the point 𝟐, 𝟎 given
that its slope 𝒎 is equal to 4.
46. Let’s Practice!
46
𝒚 = 𝟒𝒙 − 𝟖
Determine the equation of the tangent line to
the function 𝒇 𝒙 = 𝒙𝟐
− 𝟒 at the point 𝟐, 𝟎 given
that its slope 𝒎 is equal to 4.
47. Try It!
47
47
Determine the equation of the tangent
line to the function 𝒇 𝒙 = 𝟗 − 𝒙𝟐
at the
point (𝟑, 𝟎) given that its slope 𝒎 is
equal to −𝟔.
49. Let’s Practice!
49
𝒚 = −𝟏𝟏𝒙 − 𝟑𝟒
Determine the equation of the tangent line to the
function 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 − 𝟏𝟖 at the point (−𝟒, 𝟏𝟎).
50. Try It!
50
50
Determine the equation of the tangent
line to the function 𝒇 𝒙 = 𝒙𝟐
− 𝟑𝒙 − 𝟒 at
the point (𝟓, 𝟔).
51. Let’s Practice!
51
Determine the equation of the tangent line to
the function 𝒇 𝒙 =
𝟒
𝒙−𝟏
at the point (𝟑, 𝟐) given
that its slope 𝒎 is equal to −𝟏.
52. Let’s Practice!
52
𝒚 = −𝒙 + 𝟓
Determine the equation of the tangent line to
the function 𝒇 𝒙 =
𝟒
𝒙−𝟏
at the point (𝟑, 𝟐) given
that its slope 𝒎 is equal to −𝟏.
53. Try It!
53
53
Determine the equation of the tangent
line to the function 𝒇 𝒙 =
𝒙+𝟑
𝒙+𝟐
at the
point (−𝟏, 𝟐) given the its slope 𝒎 is
equal to −𝟏.
54. Check Your Understanding
54
For each item, write two expressions for the slope of
the tangent line to the given function at the given
point of tangency. No need to simplify the expression.
1. 𝑔(𝑥); (2, 4)
2. 𝑓 𝑥 = 𝑥2 − 16; (5, 9)
3. 𝑓 𝑥 = 3𝑥3
; (2, 24)
4. 𝑓 𝑥 = 𝑥2 − 𝑥; (−2, 6)
5. 𝑓 𝑥 = 𝑥2
− 2𝑥 + 3; (2, 3)
55. Check Your Understanding
55
For each item, find the slope of the tangent line at the
given point. Then, determine the equation of the
tangent line in the form 𝒚 = 𝒎𝒙 + 𝒃.
1. 𝑓 𝑥 = 3𝑥2
− 2𝑥; (−1, 5)
2. 𝑓 𝑥 =
𝑥+2
𝑥
; (1, 3)
3. 𝑓 𝑥 = 𝑥2 + 3𝑥 + 1; (1, 5)
4. 𝑓 𝑥 = −2𝑥3
; (3, −54)
5. 𝑓 𝑥 = 3𝑥2 + 1; (−2, 13)
56. Let’s Sum It Up!
56
● Equation 1: Slope of the Tangent Line
The slope of the tangent line to the function 𝑓(𝑥)
at the point (𝑎, 𝑓 𝑎 ) is given by 𝒎 = 𝐥𝐢𝐦
𝒙→𝒂
𝒇 𝒙 −𝒇(𝒂)
𝒙−𝒂
.
57. Let’s Sum It Up!
57
● Equation 2: Slope of the Tangent Line
An alternative formula for the slope of the
tangent line to the function 𝑓(𝑥) at the point
(𝑎, 𝑓 𝑎 ) is 𝒎 = 𝐥𝐢𝐦
𝒉→𝟎
𝒇 𝒂+𝒉 −𝒇(𝒂)
𝒉
where ℎ = 𝑥 − 𝑎.
58. Let’s Sum It Up!
58
● Steps in Determining the Equation of a
Tangent Line
1. Use the formula to find the slope of the
tangent line to the curve at the point of
tangency (𝑥1, 𝑦1).
2. Substitute the slope 𝑚 and the point of
tangency (𝑥1, 𝑦1) to the point-slope form of the
equation of a line: 𝒚 − 𝒚𝟏 = 𝒎 𝒙 − 𝒙𝟏 .
59. Key Formulas
59
Concept Formula Description
Equation 1:
Slope of the tangent
line to a curve
This formula gives the
slope of the tangent
line of the graph of the
function 𝑓(𝑥) at the
point (𝑎, 𝑓 𝑎 ).
𝑚 = lim
𝑥→𝑎
)
𝑓 𝑥 − 𝑓(𝑎
𝑥 − 𝑎
60. Key Formulas
60
Concept Formula Description
Equation 2:
Slope of the tangent
line to a curve
This formula gives the
slope of the tangent
line of the graph of the
function 𝑓(𝑥) at the
point (𝑎, 𝑓 𝑎 ) where
ℎ = 𝑥 − 𝑎.
𝑚 = lim
ℎ→0
𝑓 𝑎 + ℎ − 𝑓(𝑎)
ℎ
61. Key Formulas
61
Concept Formula Description
Equation of the
Tangent Line
This formula gives the
equation of the
tangent line to the
function 𝑓(𝑥) at the
point (𝑥1, 𝑦1) where 𝑚
is the slope of the
tangent line and(𝑥1, 𝑦1)
is the point of
tangency.
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
62. Challenge Yourself
62
62
A ball is thrown upward in the air. Its
height in feet is given by
𝒔 𝒕 = −𝟑𝒕𝟐
+ 𝟏𝟓𝒕 where 𝒕 is the time
in seconds. What is the
instantaneous velocity of the ball at
time 𝒕 = 𝟐 seconds?
63. Photo Credits Bibliography
63
● Slides 2 to 3: Bouncing ball strobe edit.jpg by
MichaelMaggs is licensed under CC BY-SA 3.0 via
Wikipedia.
Edwards, C.H., and David E. Penney. Calculus: Early
Transcendentals. 7th ed. Upper Saddle River, New Jersey:
Pearson/Prentice Hall, 2008.
Larson, Ron H., and Bruce H. Edwards. Calculus. 9th ed.
Belmont, CA: Cengage Learning, 2010.
Leithold, Louis. The Calculus 7. New York: HarperCollins
College Publ., 1997.
Smith, Robert T., and Roland B. Milton. Calculus. New York:
McGraw Hill, 2012.
Stewart, James. Calculus. Massachusetts: Cengage Learning,
2016.
