(b) Is the estimate in part (a) an over-estimate or underestimate of the actual area? Justify your conclusion.

Transcription

1 Chapter. Define ( ) f x = x x + x. (a) Estimate the area between the graph of f and the x axis on the interval [, ] using a lefthand sum with four rectangles of equal width. (b) Is the estimate in part (a) an over-estimate or underestimate of the actual area? Justify your conclusion. (c) Use a definite integral to calculate the exact area between the graph of f and the x axis on the interval [, ].

2 Chapter. Define ( ) f x = x x + x. (a) Estimate the area between the graph of f and the x axis on the interval [, ] using a left-hand sum with four rectangles of equal width. Δ x = =.5 x f ( x ) LHS = ( )(.5) =.5 (b) Is the estimate in part (a) an over-estimate or underestimate of the actual area? Justify your conclusion. Observe that f x = x 6x+ ( ) ( x x ) = + ( x ) = Since f >, for x >, the function f is increasing on (,]. Furthermore, since f () =, f is positive and increasing on [, ]. As a result, each rectangle in the left-hand sum will lie below the graph of f. Thus the area estimate is an underestimate. (c) Use a definite integral to calculate the exact area between the graph of f and the x axis on the interval [, ]. x x x dx + = x x + x = + + = 6 ( ) ( ) ( ) () () () + Δ x =.5 + f () + f(.5) + f() + f( Underestimate + Correct supporting work + Correctly integrate f + Correct limits of integration + Apply the Fundamental Theorem of Calculus + Area = 6

3 Chapter. Define ( ) f x = x + x. (a) Estimate the area between the graph of f and the x axis on the interval [, ] using a lefthand sum with four rectangles of equal width. (b) Estimate the area between the graph of f and the x axis on the interval [, ] using a righthand sum with four rectangles of equal width. (c) Use a definite integral to calculate the exact area between the graph of f and the x axis on the interval [, ].

4 Chapter. Define ( ) f x = x + x. (a) Estimate the area between the graph of f and the x axis on the interval [, ] using a left-hand sum with four rectangles of equal width. Δ x = =.75 x f ( x ) LHS = ( )(.75) 9.8 (b) Estimate the area between the graph of f and the x axis on the interval [, ] using a right-hand sum with four rectangles of equal width. Δ x = =.75 x f ( x ) RHS = ( )(.75) 7.59 (c) Use a definite integral to calculate the exact area between the graph of f and the x axis on the interval [, ]. x + x dx = x + x ( ) ( ) () () = + + = 9 + Δ x =.75 + f() + f(.75) + f(.5) + f(.5) Δ x =.75 + f (.75) + f(.5) + f(.5) + f() Correctly integrate f + Apply the Fundamental Theorem of Calculus + Correct limits of integration and area = 9

5 Chapter 5. Define f ( x) x( x ) = +. (a) Find the antiderivative of f that goes through (, ). (b) Calculate f ( x ) dx (c) Determine the area between f and the x axis on the interval [,].

6 Chapter 6. Define f ( x) x( x ) = +. (a) Find the antiderivative of f that goes through (, ). Let u = x +. Then du = x dx and xdx =.5du. ( ) = ( + ) = u.5du F x x x dx ( ) u =.5 C + = C Since F ( ) =, ( x ) (( ) ) ( ) = C = C =.5+ C C =.5 The specific antiderivative is F( x) ( x ) (b) Calculate f ( x ) dx ( ).5( ) F x = x + + C ( ) ( ) f ( x ) dx = F F ( ) =.65. =.75 = u (.5du) + Initial condition + ( ) ( ) F x =.5 x Fundamental Theorem of Calculus + Correct limits +.75 (c) Determine the area between f and the x axis on the interval [,]. ( ) + ( ) =.5 (.) +. (.5) f x dx f x dx =.5 + Two integrals with correct integrand + Correct limits of integration + Area =.5

7 Chapter 7. The graph of a function f is shown in the figure below. It consists of two lines and a semicircle. The regions between the graph of f and the x -axis are shaded. x+ x Editor: f( x) = x + < x < x+ x (a) Write the definite integral or sum of definite integrals that measures the area of the shaded region. (b) Calculate f ( x ) dx and ( ) f x dx. (c) Determine the area of the shaded region.

8 Chapter 8. The graph of a function f is shown in the figure below. It consists of two lines and a semicircle. The regions between the graph of f and the x -axis are shaded. Editor: x+ x f( x) = x + < x < x+ x (a) Write the definite integral or sum of definite integrals that measures the area of the shaded region. ( ) + ( ) f x dx f x dx (b) Calculate ( ) and ( ) f x dx f x dx. The area of a circle of radius is A = π. Therefore, the area of a quarter circle of radius is π. The shaded region between x = and x = is a quarter circle combined with a rectangle of area. f x dx = + π. The area of the triangular region Therefore, ( ) x = is.5. Thus ( ) between x = and f x dx =.5. f ( x) dx = + π The area of the triangular region between x = and x = is.5. However, since this region is below the x axis, f ( x) dx =.5. (c) Determine the area of the shaded region. ( ) ( ) Area = f x dx + f x dx ( ) ( ) = f x dx+ f x dx = 5 + π limits of integration + integrand + appropriately deal with the canceling of areas + Area of semi circle is π f x dx = + π ( ) + ( ) + ( ) f x dx =.5 f x dx = 5+ π + Appropriately deal with signed area + Area =.78

9 Chapter 9 5. Define ( ) f x = x x. (a) Calculate f ( x) dx. (b) What is the average value of f on [, ]? (c) What is the area of the region(s) bounded by the graph of f and the x axis? Show the work that leads to your conclusion?

10 Chapter 5. Define ( ) f x = x x. (a) Calculate f ( x) dx. f( x) dx = x x ( 6 8) ( ) = = 8 (b) What is the average value of f on [, ]? f ( x) dx avg value = 8 = = (c) What is the area of the region(s) bounded by the graph of f and the x axis? Show the work that leads to your conclusion? The x-intercepts of f are x =,,. Area = f ( x) dx + f ( x) dx = + = + F( x) = x x + Fundamental Theorem of Calculus + 8 b f ( x) dx a + b a + Correct values for a and b + + Two integrals + Correct limits of integration + Area =

11 Chapter 6. Define f( x) = x + ( x + x). (a) Use a change of variable to rewrite the integrand of function. x + dx as an easily integrable ( x + x) (b) Use integration by substitution to evaluate f ( x) dx. (c) What is the average value of f on [, ]?

12 Chapter 6. Define f( x) = x + ( x + x). (a) Use a change of variable to rewrite the integrand of x + dx as an easily integrable function. x + x ( ) Let u = x + x. Then du = x +. The integral may be rewritten as du. u u = x + x + + du u (b) Use integration by substitution to evaluate f ( x) dx. Since u = x + x, we need to determine the values of u that correspond with the limits of integration x = and x =. () () ( ) ( ) u = + = 5 u = + = Upper limit u = 7 + Lower limit u = 5 + Use integrand + or u du = u =.6 5 = b (c) What is the average value of f on [, ]? f ( xdx ) a f ( x) dx + b a avg value = + Correct limits of integration +.85 = 5.85 u

13 Chapter 7. Define f( x) =. x (a) Use the trapezoidal rule with n = 5 to estimate f ( xdx ). (b) f ( x) = ln x + C. Use the Fundamental Theorem of Calculus to calculate the exact value of f ( xdx ). (c) Explain why the trapezoidal rule cannot be used to estimate f ( xdx ).

14 Chapter 7. Define f( x) =. x (a) Use the trapezoidal rule with n = 5 to estimate b a = =. n 5 ( ) f ( xdx ). f( x) dx (b) f ( x) = ln x + C. Use the Fundamental Theorem of Calculus to calculate the exact value of f( x) dx = ln ln f ( xdx ) = ln (c) Explain why the trapezoidal rule cannot be used to estimate The function f ( xdx ). f( x) = is discontinuous at x =. Since the x,, the trapezoidal rule function f is not continuous on [ ] may not be used. b a + =. n + Proper use of trapezoidal rule Use Fundamental Theorem of Calculus + ln or.69 + Exact value + f discontinuous at x = + continuity required for trapezoidal rule

15 Chapter 5 8. A continuous, integrable function f has exactly two x-intercepts, ( x, ) and (,) (a) What is the difference in meaning between x f ( x) dx and f ( x) dx? (b) Write an integral for the area bounded by f and the x-axis. x x. F x = f x dx, (c) Given that ( ) ( ) x f( x) dx =, and ( ) x F x =, determine ( ) 5 F x.

16 Chapter 6 8. A continuous, integrable function f has exactly two x-intercepts, ( x, ) and (,) x. (a) What is the difference in meaning between x x f ( x) dx? x f ( x) dx and f ( x) dx is the signed area of the region bounded by f and the x-axis. f ( x) dx is the family of functions with derivative f x + x f ( x) dx explanation ( if x state area not signed area) explanation + f ( x) dx (b) Write an integral for the area bounded by f and the x- axis. Since the function f is continuous and has exactly two x intercepts, the area of the region bounded by f and the x- axis is given by x x F x f ( x) dx = f x dx, (c) Given that ( ) ( ) F( x ) = 5, determine ( ) ( ) ( ) ( ) x F x = F x + f( x) dx = 5+ = x F x. x x f( x) dx =, and + limits of integration + integrand f or f + integrand f + ( ) ( ) x F x F x f( x) dx + F( x ) = = + x

17 Chapter 7 9. Let graph of a continuous, twice-differentiable function f is shown in the figure below. The three regions between the graph of f and the x -axis are marked A, B, and C and have areas 5.5, 8, and 5.5, respectively. The function F is an antiderivative of f with the property that F () = 9. (a) Which value is larger F () or F ()? Justify your answer. (b) How many times does F equal 5 on the interval [, ]? Show the work that leads to your conclusion. (c) On what interval(s) is F increasing? Justify your answer.

18 Chapter 8 9. The graph of a continuous, twice-differentiable function f is shown in the figure below. The three regions between the graph of f and the x -axis are marked A, B, and C and have areas 5.5, 8, and 5.5, respectively. The function F is an antiderivative of f with the property that F () = 9. (a) Which value is larger F () or F ()? Justify your answer. () = ( ) + ( ) F F f x dx ( ) = ( ) ( ) = 9 ( 5.5) F F f x dx =.5 F is larger. ( ) ( ) = ( ) + ( ) = 9 + ( ) F F f x dx = = 6.5 (b) How many times does F equal 5 on the interval [, ]? Show the work that leads to your conclusion. We know F ( ) =.5, F () = 9, and ( ) calculate F F f x dx ( ) F = 6.5. We () = () + ( ) = =. Since F() > 5 > F(), Fc ( ) = 5 for some c in ( ) F( ) < 5 < F(), Fc ( ) = 5 for some c in ( ) function F equals 5 two times on the interval [, ].,. Since,. So the (c) On what interval(s) is F increasing? Justify your answer. When f >, F is increasing. Since f > on (,], F is increasing on (,]. b + Use ( ) ( ) ( ) F b F a f x dx = + + Use appropriate limits of integration in each integral + Conclusion with correct justification + F () = + Correct supporting work + Two times + When f >, F is increasing. + f > on (,] + F is increasing on (,]. a

19 Chapter 9. The graph of a continuous, twice-differentiable function f is shown in the figure below. The three regions between the graph of f and the x -axis are marked A, B, and C and have areas,, and 6.5, respectively. The function F is an antiderivative of f with the property that F () = 6. (a) Which value is larger F () or F (5)? Justify your answer. (b) How many times does F equal on the interval [,5 ]? Show the work that leads to your conclusion. (c) On what interval(s) is F increasing? Justify your answer.

20 Chapter. Let graph of a continuous, twice-differentiable function f is shown in the figure below. The three regions between the graph of f and the x -axis are marked A, B, and C and have areas,, and 6.5, respectively. The function F is an antiderivative of f with the property that F () = 6. (a) Which value is larger F () or F (5)? Justify your answer. 5 ( ) = ( ) + ( ) F( 5) = F( ) + ( ) f x dx = 6 = 6+ ( + 6.5) F F f x dx = F() is larger. =.5 (b) How many times does F equal on the interval [,5 ]? Show the work that leads to your conclusion. We know () 6 F =, and F (5) =.5. We F =, ( ) calculate ( ) Since F( ) Since F( ) Since F( ) F() = F() + f( x) dx = = 6. Fc = for some c in ( ) > > F(), ( ),. < < F(), Fc ( ) = for some c in (, ). 5 < < F(), Fc ( ) = for some c in (,5 ). So the function F equals three times on the interval [,5 ] because of the Intermediate Value Theorem. (c) On what interval(s) is F increasing? Justify your answer. When f >, F is increasing. Since f > on the,. interval (, ), F is increasing on ( ) b + Use ( ) ( ) ( ) F b F a f x dx = + + Use appropriate limits of integration in each integral + Conclusion with correct justification + F ( ) = 6 + Correct supporting work + Three times + When f >, F is increasing. +, f > on ( ) + F is increasing on (, ). a