Lesson 2-7

Tangent, Velocity and Rates of Change Revisited

Objectives

• Identify the average and instantaneous rates of change

Vocabulary

• Average rate of change – ∆y/∆x (the slope of the secant line between two points on the curve)

• Instantaneous rate of change – lim ∆y/∆x (∆x→0) (the slope of the tangent line at a point on the curve)

Average Rate of Change

Average Rate of Change – slope of the secant line

P

Q

a

f(a)

f(a+h)

h b

∆y f(a+h) – f(a)ms = ----- = ----------------- ∆x h

secanth = b - a

Instantaneous Rate of Change

Instantaneous Rate of Change– slope of the tangent line (derivative of the function evaluated at the point)

∆y f(a+h) – f(a)mt = lim ----- = lim ----------------- ∆x h∆x→0 h→0

P

Q

a

f(a)

f(a+h)

h b

mt

As Q gets closer and closer to P,h (∆x) gets closer to 0; and theslope of the secant line, ms, approaches the slope of the tangent line, mt

ms

Average vs Instantaneous Rate of ChangeAverage Rate of Change – slope of the secant line

Instantaneous Rate of Change – slope of the tangent line (derivative of the function evaluated at the point)

P

Q

a

f(a)

f(a+h)

h b

P

Q

a

f(a)

f(a+h)

h b

∆y f(a+h) – f(a)ms = ----- = ----------------- ∆x h

∆y f(a+h) – f(a)mt = lim ----- = lim ----------------- ∆x h∆x→0 h→0

mt

As Q gets closer and closer to P,h (∆x) gets closer to 0; and theslope of the secant line, ms, approachesthe slope of the tangent line, mt

Example 1 Find the slope of the tangent line to the curve

f(x) = 3 – 2x – 2x2 where x = 1.

323

175lim

23

3

xx

xxx

f(a+h) – f(a) lim --------------------- = h0 h

f(1+h) – f(1) lim --------------------- = h0 h

f(1) = 3 - 2 - 2 = -1 f(1+h) = 3 - 2(1+h) - 2(1+h)² = 3 - 2 - 2h - 2 - 4h -2h² = -1 - 6h - 2h²

(-1 - 6h - 2h²) – (-1) lim --------------------------- = h0 h

-6h -2h² lim ------------- = h0 h

lim -6 – 2h = -6 h0

Example 2 Find the slope of the tangent line (using the definition)

where f(x) = 1/(x-1) and a = 2.

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175lim

23

3

xx

xxx

f(a+h) – f(a) lim --------------------- = h0 h

f(2+h) – f(2) lim --------------------- = h0 h

f(2) = 1/(2-1) = 1 f(2+h) = 1/(2+h – 1) = 1/(h + 1)

(1/(h+1)) – (1) lim --------------------- = h0 h

- h / (h + 1) lim ---------------- = h0 h

- 1 lim ---------- = - 1 h0 h + 1

1 h + 1 - h------ - --------- = ----------- h + 1 h + 1 h + 1

Example 3 Find the instantaneous velocity at time t = 3 seconds if the

particle’s position at time is given by f(t) = t2 + 2t ft.

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175lim

23

3

xx

xxx

f(a+h) – f(a) lim --------------------- = h0 h

f(3+h) – f(3) lim --------------------- = h0 h

f(3) = 9 + 6 = 15 f(3+h) = (3+h)² + 2(3+h) = h² + 6h + 9 + 6 + 2h = h² + 8h + 15

(h² + 8h + 15) – (15) lim --------------------------- = h0 h

h² + 8h lim ------------- = h0 h

lim h + 8 = 8 h0

Example 4 For a particle whose position at time t is f(t) = 6t2 - 4t +1 ft.: 

a. Find the average velocity over the following intervals:

 

i. [1, 4]

ii. [1, 2]

iii.[1, 1.2]

iv.[1, 1.01]

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175lim

23

3

xx

xxx

∆yAve vel = ms = --------- ∆x

f(4) – f(1) 81 – 3 78---------------- = ------------- = -------- = 26 4 – 1 3 3

f(2) – f(1) 17 - 3---------------- = ------------ = 14 2 – 1 1

f(1.2) – f(1) 4.84 - 3 1.84------------------ = -------------- = -------------- = 9.2 1.2 – 1 0.2 0.2

f(1.01) – f(1) 3.0806 - 3 .0806-------------------- = --------------- = ------------- = 8.06 1.01 – 1 .01 .01

Example 4 continued For a particle whose position at time t is f(t) = 6t2 - 4t +1 ft.:

 

b. Find the instantaneous velocity of the particle at t = 1 sec.  

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175lim

23

3

xx

xxx

f(a+h) – f(a) lim --------------------- = h0 h

f(1+h) – f(1) lim --------------------- = h0 h

f(1) = 6 - 4 + 1 = 3 f(1+h) = 6(1+h)² - 4(1+h) + 1 = 6h² + 12h + 6 – 4h – 4 + 1 = 6h² + 8h + 3

(6h² + 8h + 3) – (3) lim --------------------------- = h0 h

6h² + 8h lim ------------- = h0 h

lim 6h + 8 = 8 h0

Summary & Homework

• Summary:– Average rate of change is the slope of the

secant line– Instantaneous rate of change is the slope of

the tangent line

– Limit of m secant = m tangent as ∆x →0

• Homework: pg 155 - 157: 2, 7, 15, 22, 27