1.2 The Slope of the Tangent A Lines The slope of a line - La Citadelle

If P ( a,f ( a))and Q ( a + h,f ( a + h))then the slope ofthe secant line is given by:f ( a + h)− f ( a)m =hD Tangent LineAs the point Q approaches the point P , the secantline approaches the tangent line at P . See thediagram on the right side.The slope of the tangent line at P( a,f ( a))is:Calculus and Vectors – How to get an A+f ( a + h)−m = limh→0 hf ( a)(1)E Graphical Computation1. Draw the tangent line using a ruler2. Choose two points on the tangent line A ( x 1 , y1)and B ( x 2 , y2)3. Use the formula:y2− y1m ≅x2− x1to estimate the slope of the tangent line.Ex 4. Find the slope of the tangent line to the curvey = 2 x −1 at the point P (5,4).Note. The slope of the tangent line can becalculated using:m = tanαwhere α is the angle between the tangent line andthe positive direction of the x-axis.F Numerical Computationf ( a + h)− f ( a)1. Use the formula m ≅(2)h2. Choose a sequence h 1, h2, h3, ... → 0 .3. Compute m 1, m2, m3,.. .4. Observe the pattern and conclude.5. Be careful at “difference catastrophe”.3Ex 5. Consider y = f ( x)= x . Estimate numerically the slopeof the tangent line at P (1,1 ) using−201 = 0.1, h2= 0.0001, h3= 10h .Note. The difference catastrophe is related to thelimited capacity of memorizing numbers by anytechnologic device (scientific calculator, computer,etc). If two numbers are very close, then they havethe same internal representation in the memory.1.2 The Slope of the Tangent - Handout©2010 Iulia & Teodoru Gugoiu - Page 2 of 3

Calculus and Vectors – How to get an A+G Algebraic Computationf ( a + h)− f ( a)1. Use the formula m = lim.h→0h2. Do not substitute h by 0 because you will getthe indeterminate case 00 .3. Compute algebraic the difference quotientf ( a + h)− f ( a)DQ = until you succeed to cancelhout the factor h .4. Substitute in the remaining expression h by 0 .Ex 6. Find the slope of the tangent line to the graph ofy = f ( x)= x2 − 3xat the point P ( 1, −2).Ex 7. Find the equation of the tangent line to the1graph of y = f ( x)= at the point P (2,1).x −1Ex 8. Consider2 −y = f ( x)= x 2x.a) Find the slope of the tangent line at the generic pointP ( a,f ( a)).b) Find the point where the tangent line is horizontal.c) Find the point P such that m = 2 .Pd) Find the point P such that the tangent line at P isperpendicular to the line L : x − 3y3 .2 =Reading: Nelson Textbook, Pages 10-18Homework: Nelson Textbook: Page 16, #1a, 2a, 3ac, 4a, 5a, 6a, 8a, 9a, 10a, 11a, 15, 21, 251.2 The Slope of the Tangent - Handout©2010 Iulia & Teodoru Gugoiu - Page 3 of 3