BCC.01.2 - A Closer Look at Rates of Change: The Tangent Problem

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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

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### 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
• Instantaneous Rate of Change at x1 can be estimated by making the interval x2 - x1 smaller and smaller which we can present as a tangent line to the curve at the point x1
(B) Tangent Slopes
• 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 xa
• The process outlined in the previous slide is animated for us in the following internet links:
• Secants and tangent
• A Secant to Tangent Applet from David Eck
• JCM Applet: SecantTangent
• Visual Calculus - Tangent Lines from Visual Calculus – Follow the link for the Discussion
(D) Tangent Slope Equations
• We can put the last two ideas together in a special notation 
(E) Limits of Tangent Slopes
• To interpret these statements, we are looking at a sequence of secant slopes (y/x) that happen to reach some limiting slope value. The limiting slope value we reach is the tangent slope.
• To see this idea from a table of values generated from the limit equation given in the previous slide, follow the link below:
• Visual Calculus - Slopes of Tangent Lines
(F) Example #1
• ex 1. Let f(x) = ½ x² and we will find the slope of the tangent at the point (1,1). We will set it up by working through the following table:
• we can make use of the previous link from Visual Calculus
• Or we can use the secant formula as follows:
(G) Example #2
• 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
• Thus y = 11x + 2 would be our tangent slope
(H) Example #3
• 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