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

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

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

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

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 =

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

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 =

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

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

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

Now lets make it a tangent ·

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

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.

Whew!!!! That Was Easy!