1. Welcome To Calculus (Do not be afraid, for I am with you)
The slope of a tangent line

2. Questions To Think About
If you drive 150 miles in 3 hours, what’s your average speed?
50 mph is your AVERAGE RATE OF CHANGE
What is the general maths term for AVERAGE RATE OF CHANGE?
SLOPE of a line = AVERAGE RATE OF CHANGE
Did you really drive 50 mph constantly on your journey?
NO…that was your AVERAGE RATE OF CHANGE
Fill in the blank: When driving, I looked at my speedometer and it read 65 mph. At this instant, 65 mph was my _______________________ rate of change.
INSTANTANEOUS

3. Pretty Fast (Got stopped by the police about 15 minutes later)
Tell me something about your INSTANTANEOUS RATE OF CHANGE 1/2 hour into the trip.
Pretty Fast (Got stopped by the police about 15 minutes later)
Tell me something about your INSTANTANEOUS RATE OF CHANGE 2 hours into the trip.
Went backwards because I was lost.
150
150 miles in 3 hours = AVE RATE of 50 mph
Distance In Miles 1 2 3
Time In Hours

4. The Slope Of The Red Line
Let’s find the slope of the tangent line to y = x2 when x = 2 (Instantaneous Rate of Change)
The Slope Of The Red Line

5. We begin by setting up what’s called a secant line through (2,4) and (2+h,(2+h)2)
The slope of that line =

6. As h gets closer and closer to zero, we approach our tangent line
As h gets smaller, the secant line approaches the tangent line, and the average rate of change becomes the instantaneous rate of change h h
As h gets closer and closer to zero, we approach our tangent line

7. When h approaches zero, our slope equation becomes….
(2+h,(2+h)2)
(2,4) h
(2+h)2 - 4
The slope of tangent line =

8. Lets Evaluate The Limit= = = 4 =
The Slope of the Tangent Line at (2,4) = 4

9. · y = f(x) (x+h,f(x+h)) (x+h,f(x+h)) (x,f(x)) CLICK TO CONTINUE

10. Find the slope of the tangent line for any point (x,f(x)) for f(x)=x2
Start with the slope of a secant

11. Now lets make it a tangent

12. We say that the derivative of is 2x
The formula that will calculate the slope of a tangent line for at any point (x,y) is 2x 6
For example:
At (3,9) the slope of the tangent is … -8
At (-4,16) the slope of the tangent is …
We say that the derivative of is 2x

13. SUMMARY
To find the equation of the tangent line, simply find f ’(a), that is your slope. Now use your point of tangency {(a,f(a)} and your slope, m = f ‘(a) in the point – slope form of a linear equation.

14. Whew!!!!
That Was Easy!