Differential Calculus, limits, slope of tangent line to the curve, the derivative. This slide accompanies my lecture in Differential calculus in LPU Batangas

1. {
The Derivative
Slope of a line, secant line, tangent line, slope of a
secant line, slope of a line tangent to a curve

2. Lines, slope, point-slope
form

3. What do we know about slopes of lines?
Lines, slope, point-slope
form

4. Lines, slope, point-slope
form

5. Lines, slope, point-slope
form

6. Lines, slope, point-slope
form

7.  Given a curve, the line that
intersects the curve on two points is
called a secant line.
Secant line

8.  Given a curve, the line that
intersects the curve on two points is
called a secant line.
Secant line

9. Tangent line

10.  Given a non-linear curve, how do you get the
slope of the curve at a the point (x, f(x) )?
Challenge question

11.  Given a non-linear curve, how do you get the
slope of the curve at a the point (x, f(x) )?
Challenge question
Let 𝑚 𝑇𝐿 be the slope
of the tangent line,
then the slope of the
curve at (x,f(x)) is the
same as 𝑚 𝑇𝐿
𝑚 𝑇𝐿 is the derivative
of f at x.
𝑓′ 𝑥
= 𝑚 𝑇𝐿
𝑦 = 𝑓(𝑥)

12.  How do we compute 𝑚 𝑇𝐿?
Challenge question

13. How to find 𝑚 𝑇𝐿
𝑥𝑖, 𝑓 𝑥𝑖 𝑚 𝑇𝐿 =
Δ𝑦
Δ𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥2
𝑓′ 𝑥 = lim
Δ𝑥→0
𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥
∆𝑥

14.  Three step rule
1. Find 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥
2. Find
𝑓 𝑥+∆𝑥 −𝑓 𝑥
∆𝑥
1. Evaluate
How to compute for 𝑓′ 𝑥
lim
Δ𝑥→0
𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥
∆𝑥

15. How to compute for 𝑓′(𝑥)
First step

16. How to compute for 𝑓′(𝑥)
Second step

17. How to compute for 𝑓′(𝑥)
Third step
𝑓′ 𝑥 = 4𝑥 − 3

18. What is 𝑓′ 2 ?
𝑓′ 2 = 4 2 − 3 = 5
What is 𝑓′
0 ?
𝑓′ 0 = 4 0 − 3 = −3