CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

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1 CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental Theorem of Calculus (Chain Rule Version): d g f tdt d a E. Use the Second Fundamental Theorem to evaluate: d (a) t dt d 3 d 3 (b) t d tan dt (c) d d 3 dt t (d) d d sin 3 t dt

2 E. The graph of a function f consists of a quarter circle and line segments. Let g be the function given by g f t dt. (a) Find g, g, g, g 5. y Graph of f (b) Find all values of on the open interval, 5 at which g has a relative maimum. (c) Find the absolute minimum value of g on, 5 and the value of at which it occurs. (d) Find the -coordinate of each point of inflection of the graph of g on, 5. Justify your answer.

3 CALCULUS WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND FUNCTIONS DEFINED BY INTEGRALS. Find the derivatives of the functions defined by the following integrals: sin t cos (a) dt t (b) e dt t (c) dt t (d) tan t e dt (e) (f) cos t dt, t dt (g) s cos 3 ds (h) cos s 5 t t dt 7 4 (i) sin t dt tan. The graph of a function f consists of a semicircle and two line segments as shown. Let g be the function given by g f tdt. (a) Find g g g g, 3,,and 5. (b) Find all values of on the open interval,5 at which g has a relative maimum. Justify your answers. (c) Find the absolute minimum value of g on the closed interval [,5] and the value of at which it occurs. (d) Write an equation for the line tangent to the graph of g at = 3. (e) Find the -coordinate of each point of inflection of the graph of g on the open interval,5. (f) Find the range of g.

4 3. Let g f t dt, where f is the function whose graph is shown. g, g, g,and g 6. (a) Evaluate (b) On what intervals is g increasing? (c) Where does g have a maimum value? What is the maimum value? (d) Where does g have a minimum value? What is the minimum value? (e) Sketch a rough graph of g on [, 7]. 4. Let g f t dt, where f is the function 3 whose graph is shown. g 3 and g 3. (a) Evaluate (b) At what values of is g increasing? Justify. (c) At what values of does g have a maimum value? Justify. (d) At what values of does g have a minimum value? Justify. (e) At what values of does g have an inflection point? Justify.

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6 CALCULUS WORKSHEET ON FUNCTIONS DEFINED BY INTEGRALS. Find the equation of the tangent line to the curve y F where at the point on the curve where =. 3 F t 7 dt 3. Suppose that 5 4 (a) What is f? c f t dt. (b) Find the value of c. 3. If 3 F t t dt, for what values of is F decreasing? 4 4. Let H f t dt where f is the continuous function with domain [, ] shown on the right. (a) Find H. y (b) On what interval(s) of is H increasing? Graph of f (c) On what interval(s) of is H concave up? (d) Is H positive or negative? Eplain. (e) For what value of does H achieve its maimum value? Eplain.

7 5. The graph of a function f consists of a semicircle and two line segments as shown on the right. Let g f t dt. (a) Find g, g 3, g. y (b) On what interval(s) of is g decreasing? Justify your answer. Graph of f (c) Find all values of on the open interval 3, 4 at which g has a relative minimum. Justify your answer. (d) Find the absolute maimum value of g on the interval 3, 4 and the value of at which it occurs. (e) On what interval(s) of is g concave up? (f) For what value(s) of does the graph of g have an inflection point? (g) Write an equation for the line tangent to the graph of g at. 6. The graph of the function f, consisting of three line segments, is shown on the right. Let g f t dt. (a) Find g, g 4, g. y. (b) Find g and g 3 (c) Find the instantaneous rate of change of g with respect to at =. Graph of f (d) Find the absolute maimum value of g on the interval, 4. (e) The second derivative of g is not defined at = and at =. Which of these values are -coordinates of points of inflection of the graph of g?

8 CALCULUS WORKSHEET 3 ON FUNCTIONS DEFINED BY INTEGRALS Work the following on notebook paper.. The function g is defined on the interval [, 6] by g f t dt where f is the function graphed in the figure. (a) For what values of, < < 6, does g have a relative maimum? (b) For what values of is the graph of g concave down? (c) Write an equation for the tangent line to g at the point where = 3. (d) Sketch a graph of the function g. List the coordinates of all critical point and inflection points. 3 f 7. Find the. Suppose that f is a continuous function, that f, and that average value of f over the interval [, ]. 3. The graph of a differentiable function f on the closed interval [ 4, 4] is shown. Let G f t dt for (a) Find G 4. (b) Find G 4. (c) On which interval or intervals is the graph of G decreasing? (d) On which interval or intervals is the graph of G concave down? (e) For what values of does G have an inflection point?

9 4. The function F is defined for all by (a) Find F. 8. F t dt (b) Find F. (c) Find F. (d) Find F If 6 F t t dt, on what intervals is F decreasing? 6. The graph of the velocity vt, in ft/sec, of a car traveling on a straight road, for t 35, is shown in the figure. (a) Find the average acceleration of the car, in ft / sec, over the interval t 35. (b) Find an approimation for the acceleration of the car, in ft / sec, at t =. Show your computations. 35 (c) Approimate v t dt with a Riemann sum, using 5 the midpoints of three subintervals of equal length. Eplain the meaning of this integral.

10 7. The function F is defined for all by F f t dt, where f is the function graphed in the figure. The graph of f is made up of straight lines and a semicircle. (a) For what values of is F decreasing? (b) For what values of does F have a local maimum? A local minimum? F, F, and F. (c) Evaluate (d) Write an equation of the line tangent to the graph of F at = 4. (e) For what values of does F have an inflection point?

11 Answers to Worksheets on Second Fund. Th. & Functions Defined by Integrals. (a) sin e (b) (c) - tan (d) (e) (f) cos (g) (h) sin cos cos cos 3 (i) sin tan 4 sec. (a),,, (b) g has a rel. ma. at = because g f changes from positive to negative there. (c) Abs. min. = at = - (Justify with Candidates Test.) (d) y 3 (e) g has an I.P at = because g changes from increasing to decreasing there. g has an I.P at = 3 because g changes from decreasing to increasing there. (f) [, ] 3. (a),, 5, 3 (b) g is increasing on (, 3) since g is positive there. (c) Ma. value = 7 at = 3 (Justify with Candidates Test.) (d) Min. value = at = (Justify with Candidates Test.) 3 (e) y 4 4. (a) g is decreasing on, and (4, 5) because g f is negative there. g f changes from positive (b) g has a rel. ma. at = and at = 4 because to negative there. 3 (c) g is concave down on, 4 because g f is decreasing there. 3 (d) g has an I.P at,, and 3 because g changes from increasing to 4 4 decreasing or vice versa there. (Answers to 5 & 6 on net page)

12 5. (a) g() table values:, -, -, -3, -3.5, -3, -.5,.5,.5,.5,.5 (b) plot (c) g has minimum at = 4 b/c g (4)= and changes from negative to positive there (d) four points on g that are collinear are at =,, and 3 (e) g increases at greatest rate between = 6 and = 7 because integral from 6 to 7 is greates 6. (a) F has critical numbers at =,,4,6,8, because F () = there (b) F is decreasing on (,4) and (6,8) because F () < there (c) F is concave up on (,),(3,5) & (7,9) because F () > there (or F () is increasing there) Worksheet on Functions Defined by Integrals. y. (a) 5 (b) there. 3. F is decreasing on < 3 because F 4. (a) (b) H is increasing on (, 6) because H f is positive there. (c) H is concave up on (9.5, ) because H f is increasing there. (d) H is positive because there is more area above the -ais than below. (e) H achieves its maimum value at = 6 because H and H 6 and H are positive and H 6 H. 5. (a), -, (b) g is decreasing on (, 3) because g f is negative there. (c) g has a relative minimum at = 3 because g f changes from negative to positive there. (d) Abs. ma. = at = (Justify with Candidates Test.) 3, and, 4 g f is increasing there. (e) g is concave up on because (f) g has an inflection point at and = because g f increasing to decreasing or vice versa there. y (g) 6. (a),, (b), (c) 9 (d) Abs. ma = 5 at = 3 (Justify with Candidates Test.) changes from (e) g has an inflection point at = because g f changes from increasing to decreasing there. g does not have an inflection point at = because g f is decreasing for < < and continues to decrease on < < 4.

13 Worksheet 3 on Functions Defined by Integrals g, which is f, changes from positive to. (a) g has a rel. ma. at = because negative there., which is, is decreasing (b) g is concave down on (, 3) and (5, 6) because g f there. y 3 (d) graph (c) (a) (b) is negative there. (c) G is decreasing on (, 3) because G, which is f, (d) G has a rel. min. at = 3 because G f positive there., which is, changes from negative to, which is, is (e) G is concave down on 4, 3 and, because G f decreasing there. (f) G has an inflection point at 3,, and 4. (a) changes from decreasing to increasing or vice versa there. 4 8 (b) 6 because G f, which is, (c) 8 (d) F is decreasing on and 3 because F is negative there (a) ft / sec 7 (b) ft / sec (using (, 4) and (5, 3) to estimate the slope) (c) ()(3) + ()(4) + ()() = 9 ft. This integral represents the approimate distance in feet that the car has traveled from t = 5 seconds to t = 35 seconds. 5, 3.5 and, 5 F, which is f, is negative 7. (a) F is decreasing on because there. (b) F has a local minimum at 3.5 because F, which is f, changes from negative to positive there. F has a local maimum at = because F f changes from positive to negative there. (c) 4,, 3 (d) y 4 (e) F has an inflection point at 3,,, and 3 changes from increasing to decreasing or vice versa there., which is, because F f, which is,