Transcript Document
FTC PART 2 Calculus - Santowski 7/15/2015 Calculus - Santowski 1 Review • We have seen a definition/formula for a definite integral as n b A(x) lim f x i x i1 a n f (x)dx F(x) a F(b) F(a) b • And have seen an interpretation of the definite integral as a “net/total change” • Many textbooks/resources refer to this statement as the Integral Evaluation Theorem as it tells us HOW to evaluate a definite integral by finding antiderivatives 7/15/2015 Calculus - Santowski 2 FTC, PART 1 • Many textbooks/resources refer to this statement as the Integral Evaluation Theorem as it tells us HOW to evaluate a definite integral b f (x)dx F(x) a F(b) F(a) b a by finding antiderivatives • BUT what if we can’t DETERMINE an equation for the antiderivative?? 7/15/2015 Calculus - Santowski 3 FTC, PART 1 & 2 • BUT what if we can’t FIND an expression/equation for the antiderivative?? • Consider the equations f (t) sin2t g(t) sin t 1 t • One has an “easy” antiderivative, but what about the other???? 7/15/2015 Calculus - Santowski 4 FTC, Part 2 • If f is continuous on [a,b] then the fcn F(x) has a derivative at every point in [a,b] and d d F(x) dx dx x x f (t)dt a f (t)dt f (x) a • So what does this really mean Every continuous function f(x) HAS an antiderivative (which simply happens to be expressed as an integral as: x f (t)dt ) rather than explicitly in terms of elementary a functions! 7/15/2015 Calculus - Santowski 5 FTC PART 2 • • • • To show a geometrical interpretation of FTC, Part 2 f(t) represents a curve in the t-y plane and F(x) represents the area under y=f(t) between a and some arbitrary x value and is a function of x (as x changes, so does the area) The derivative F’(x) is the rate of change of area (the dark vertical line in the diagram) 7/15/2015 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Calculus - Santowski 6 FTC: PART 2 - Example • So how does this this help us with antiderivatives?? • Again, let’s go to a geometric & graphic representation and work with finding an antiderivative for 1 1 x 7/15/2015 Calculus - Santowski 7 FTC, PART 2 - Example Qui ckTi me™ and a TIFF (Uncompressed) decompressor are needed to see this pictur e. Qui ckTi me™ and a TIFF (Uncompressed) decompressor are needed to see this pictur e. 7/15/2015 Qui ckTi me™ and a TIFF (Uncompressed) decompressor are needed to see this pictur e. • So what have we done?? We have created a GRAPH of the antiderivative rather than having developed an explicit formula in terms of elementary functions Calculus - Santowski 8 FTC Part 2 - An Illustration • The same geometric/graphic connection can be made using the following internet link: • FTC Part 2 from Visual Calculus 7/15/2015 Calculus - Santowski 9 Using the FTC • Given the function f be as shown and let g(x) x f (t)dt 0 • (a) Evaluate g(0), g(2), g(4), g(7) and g(9) and g(11) • (b) On what intervals is g increasing? • (c) Where does g have a maximum value? • (d) Sketch a rough graph of g(x) 7/15/2015 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Calculus - Santowski 10 Using the FTC, Part 2 • Find the derivative of the following functions g(x) x t 2 t dt 20 1 h(x) t x 1 3 1 dt x • For all real numbers of x, define F(x) sint dt 0 3 1 Evaluate F and F and interpret 4 2 7/15/2015 Calculus - Santowski 11 Using the FTC, Part 2 • Graph the function defined by F(x) x e t 2 dt 0 • • • • • Address the following in your solution: F’(x) (i) determine and discuss (ii) determine and discuss F’’(x) (iii) find symmetry of F(x) (iv) estimate some points using trapezoid sums 7/15/2015 Calculus - Santowski 12 Using the FTC, Part 2 x • Find the interval on which the curve is concave up 1 y dt 2 0 1 t t 7/15/2015 Calculus - Santowski 13 Using the FTC, Part 2 x 1 dt • Given that F(x) 2 0 1 t • (a) Find all critical points of F • (b)Determine the interval on which F increase and F decreases • (c) Determine the intervals of concavity and inflection points of F • (d) Sketch a graph of F 7/15/2015 Calculus - Santowski 14 FTC, Part 2 • What is the advantage of defining an antiderivative as an integral? => we can then simply use our numerical integration methods (RRAM, LRAM etc) to estimate values 7/15/2015 Calculus - Santowski 15 Homework • Handout from Stewart, 1998, §5.4, p390391, • (ii) Graphs: Q7-10, 23-24,26,28 • (ii) Algebra: Q11-16 7/15/2015 Calculus - Santowski 16