A3 – Secants and Tangents IB Math HL&SL - Santowski

(A) Review Average Rate of Change = which represents a secant line to a curve Instantaneous Rate of Change at x 1 can be estimated by making the interval x 2 - x 1 smaller and smaller  which we can present as a tangent line to the curve at the point x 1

(A) Review - Graphs

(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 x 1, x 2,......, x n as seen on the previous diagram When we move our secant point x 1, x 2,......, x n 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 x 1, x 2,......, x n gets closer and closer to our tangency point, a second note is that the interval between x n 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

(C) Internet Links - Applets 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 DiscussionVisual Calculus - Tangent Lines from Visual Calculus

(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 Calculusprevious link from Visual Calculus Or we can use the secant formula as follows: X Secant slope

(F) Example #1 - Graph

(G) Example #2 Find an equation to the tangent line to f(x) = 3x 3 + 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 m secant = (3x 3 + 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 = into our equation to estimate the tangent slope  then tangent slope = so our tangent slope estimate would be 11 Thus y = 11x + 2 would be our tangent slope

(G) Example #2 - Graph

(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 - t²) 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

(I)Internet Links Calculus I (Math 2413) - Limits - Tangent Lines and Rates of Change from Paul Dawkins at Lamar UniversityCalculus I (Math 2413) - Limits - Tangent Lines and Rates of Change from Paul Dawkins at Lamar University Limits of Functions - Applets, Explanations, Flash Limits of Functions - Applets, Explanations, Flash Tutorials

(J) Homework Stewart, 1989, Calculus – A First Course, Chap 1.4, p35, Q7(ii,iv,v,vi),8bd & Chap 1.5, p43, Q1,2,6,8