3.1 The Limit Definition of the Derivative September 25, 2015
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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.
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