Aug 9, 2011 ... Limits, derivatives and integrals. Limits and Motion: The Tangent Problem.
Average Velocity is the change in position divided by the change in.
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Average Velocity is the change in position divided by the change in time.
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
So Galileo did the best he could by making Δt as small as experimentally possible, measuring the small values of Δs, and then finding the quotients. It only approximated the instantaneous velocity, but finding the exact value appeared to be algebraically out of the question, since division by zero was impossible.
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem Let's review what slope is... What is the formula? What does it mean?
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
We can calculate the slope of the green line easily if we know the coordinates of P and Q...
P
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
...however, it is only truly the slope of this curve if it is TANGENT to the curve.
P
The question now is how to we make the line tangent?
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
Does it make sense that if we move P and Q closer together along the curve, that the line will eventually become a tangent line?
P
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
f(a +h )
Now, we should be able to calculate the average rate of change for the time between a and a + h.
P f (a )
a
h
a + h
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
f(a+h) Now, looking at the line between the two points, we can calculate average velocity, right?
P
Using the same slope formula y2 y1 x2 x1 what is y2? y1? x2? x1?
f(a)
a
h
a + h
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
f(a+h) So...
P f(a)
a + h
h
a
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
f(a+h) But don't we want P and Q to meet? How does that happen? What can we do?
P f(a)
a
h
a + h
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem y
y = f(x)
Q
f(a+h) YES! YES! YES! YES!
h approaches zero!!!
P
Therefore...
f(a)
a
h
a + h
x
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
...how could we evaluate h approaching zero?
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem The slope of a curve at a given point is known as the derivative of the curve.
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
Chapter 10 ‐ An introduction to calculus: Limits, derivatives and integrals Limits and Motion: The Tangent Problem
