BCC.01.2 - A Closer Look at Rates of Change: The Tangent Problem. MCB4U - Santowski. (A) Review. Average Rate of Change = which represents a secant line to a curve
PowerPoint Slideshow about 'BCC.01.2 - A Closer Look at Rates of Change: The Tangent Problem' - sachi
An Image/Link below is provided (as is) to download presentation
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
We can estimate tangent slopes at a given point x = a (the tangency point) from secant slopes by drawing and determining the slopes of secants lines from point x1, x2, ......, xn as seen on the previous diagram
When we move our secant point x1, x2, ......, xn closer and closer to our tangency point at x = a, we create a sequence of secant slopes which we can use as estimates of our tangent slope is there an end or a limit to the secant slopes??? The answer is yes.
As our secant point x1, x2, ......, xn gets closer and closer to our tangency point, a second note is that the interval between xn and x = a (which we can call x) gets smaller and smaller and approaches 0.
We can write that as x 0 or we can also present as it xa
Find an equation to the tangent line to f(x) = 3x3 + 2x - 4 at the point (-1,-9).
An alternative approach to this solution is that we can generate an equation to help us out slope of EVERY secant =
So the rational equation msecant = (3x3 + 2x + 5)/(x+1) will generate for us every slope of every secant that we are wanting to calculate we simply input the secant point’s x co-ordinate and thereby generate the secant slope between our 2 points
To simplify the process, given our previous observations from Example #1, let’s substitute in x = -1.0001 into our equation to estimate the tangent slope then tangent slope = 11.0009 so our tangent slope estimate would be 11
ex 3. Dwayne drains a tub which holds 1600 L of water. It takes 2 h for the tub to drain completely. The volume of water remaining in the tub is V(t) = 1/9(120 – t)² where V is volume in litres and t is time in minutes.
(a) Determine the average rate of change during the second hour
(b) Determine the instantaneous rate of change after exactly 60 minutes