3.1 The Limit Definition of the

Derivative September 25, 2015

ObjectivesO I can find the derivative of a

function using the limit definition of a derivative

O I can evaluate the slope of a curve (the derivative) at a specific point on the curve

O I can write the equation of a line tangent to a curve at a certain point

Agenda O Discussion of Unit 2 Limits Exam (5

minutes) O Lesson Warm-Up (20 minutes) O Notes on the Limit Definition of a

Derivative with built-in Guided Practice (39 minutes)

O In-class Practice Time (20 minutes)O Exit Ticket (10 minutes)

Lesson Warm-Up (10 min.)1. Find the slope of the line that connects the

two points P( 4, 5 ) and Q ( -2, 3 ) 2. Write the equation of the line PQ. 3. For the function f(x) = x2 – 5, evaluate

1. f(4) = ?2. f(h) = ?3. f(x+ h) = ?

Lesson Warm-Up (10 min.)1. Find the slope of the line that connects the

two points P( 4, 5 ) and Q ( -2, 3 )

Slope formula

Lesson Warm-Up (10 min.)1. Find the slope of the line that connects the

two points P( 4, 5 ) and Q ( -2, 3 ) 2. Write the equation of the line PQ.

Lesson Warm-Up (10 min.)1. Find the slope of the line that connects the

two points P( 4, 5 ) and Q ( -2, 3 ) 2. Write the equation of the line PQ. 3. For the function f(x) = x2 – 5, evaluate

1. f(4) = ?2. f(h) = ?3. f(x+ h) = ?

f(4) = 42 – 5 = 16 – 5 = 11 f(h) = h2 – 5 f(x+h) = (x +h)2 – 5

= (x + h) (x + h) – 5= x2 + xh + xh + h2 – 5 = x2 + 2xh + h2 – 5

Rate of ChangeConsider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = 2t + 1

Where is the object at the 1st second?

t = 1 seconds(1) = 2(1) + 1 = 3 meters

Rate of ChangeConsider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = 2t + 1

Where is the object at the 2nd second?

t = 2 secondss(1) = 2(2) + 1 =5 meters

Rate of ChangeConsider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = 2t + 1

What is the rate of change ?

Rate of ChangeConsider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = 2t + 1

What is the rate of change ?

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the AVERAGE rate of change between t = 1 and t = 2 seconds?

Secant line

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

tangent line at t = 1

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

Consider: An object is moving and its position s(t) is measured in meters and depends on t

in seconds s(t) = t2

What is the INSTANTANEOUS rate of change between at exactly the FIRST second?

The Derivative The derivative of f(x) at x = a,

Finding the DerivativeExample 1: Write the equation of the line that is tangent to the curve y = x2 at the point (1, 1).

Finding the DerivativeExample 1: Write the equation of the line that is tangent to the curve y = x2 at the point (1, 1).

Step 1: Find the derivative (= slope of the curve) at the point (1, 1)

Finding the DerivativeStep 1: Find the derivative (= slope of the curve) at the point (1, 1)

Finding the Derivative

Finding the Derivative

Writing the EquationDerivative = slope of the curve = slope of

the tangent

Slope = 2 m/s

Point = (1, 1)

Writing the EquationSlope = 2 m/sPoint = (1, 1)

Finding the DerivativeExample 2: Write the equation of the line that is tangent to the curve y = x3 + x when x = 0.

Finding the Derivative

Writing the EquationSlope = 1

Point = (0, 0)

Guided Practice Problems

1. Write the equation of the line tangent to the curve f(t) = t – 2t2 at a = 3.

2. f(x) = 4 – x2 at a = -1

3. at a = 3

4. at a = -2

Homework AssignmentWrite the equation of the tangent line of the following curves at the given points. 1. f(x) = 2x2 + 10x , a = 32. f(x) = 8x3 , a = 1 3. , a = 14. , a = 0

Exit Ticket 1. Compute the derivative and write

the equation of the tangent line at a = -1 for the following function: f(x) = 3x2 + 4x + 2

2. In full sentences, explain the relationship how a secant line is different from a tangent line and how average velocity is different from instantaneous velocity.