TEST 1 FORM B MAT 168

Transcription

1 TEST FORM B MAT 68 Directions: Solve as many problems as well as you can in the blue eamination book, writing in pencil and showing all work. Put away any cell phones; the mere appearance will give a zero.. Give both a geometric and an algebraic definition of the Riemann Sum. geometric: an approimation of the area under a curve, using a finite number of rectangles algebraic: f i ) where i is any sample point in the ith subinterval of [a, b] and = b a n. Compute the following its, if they eist. Use L Hôpital s Rule only if necessary. a) As, we are dividing a constant by larger and larger positive numbers, to the quotient dwindles to zero, so the it is zero. L Hôpital s Rule does not apply. b) + As from the left, we are dividing a constant by smaller and smaller negative numbers, so the quotient decreases without bound, so the it is. L Hôpital s Rule does not apply. + sin c) + sin, so L Hôpital s Rule applies directly. Take the derivative of numerator and denominator and we have + sin L H = + cos = + =. d) ln + ), so L Hôpital s Rule applies directly. ln + ) numerator and denominator, without forgetting the chain rule, and we have Take the derivative of ln + ) L H = = 4 + ) 5 ), Date: Fall 7.

2 e) so L Hôpital s Rule applies again. We have ln + ) = ) TEST FORM B L H = L H = = 5. 9 ) 9, so L Hôpital s Rule, if it applies, applies indirectly. Multiply by the conjugate and we have = 9 9 ) + 9 = Here L Hôpital s Rule applies directly, and we have ) 9 = L H = + 8 9,. which is a right mess, but multiply numerator and denominator by 9 and we have ) 9 9 = ). This still approaches /, and even worse, it seems to grow more complicated rather than less, so we contemplate a different approach: multiply numerator and denominator by /. That gives us ) 9 9 = ) = ) = = 6. f) + ) ) 5 approaces the form, so L Hôpital s Rule, if it applies, applies indirectly. Since the variable is in the eponent, we need to use the natural + logarithm: y = + ) 5 ln y = ln + ) 5 ln y = 5 ln + ) 5 ln + ) ln y = 5 ln + ) ln y = + + Now L Hôpital s Rule apples, and we have 5 ln + ) ln y = + + L H 5 = + +. = + + =.

3 TEST FORM B This is the it of ln y, so + + ) 5 = + y = + eln y = e + ln y = e.. Suppose we want to approimate 9 by using Newton s Method to find a root of 9, starting at =. a) Find the first four approimations. We are given f ) = 9 and an initial approimation =. Newton s Method computes each successive approimation i+ using so we compute f ) = and thus i+ = i f i) f i ), = f ) f ) = 9 = 5.8 = f ) f ) = = f ) f ) = = 4 f 4) f 4 ) = b) Are they correct to the nearest thousandth place? Why or why not? The final approimation should be correct because the Method has repeated in the thousandths place three times now. 4. We want to find or approimate a) A = d. Use high school geometry to approimate the area under the curve. To be clear: I do not want anything sophisticated here. I do not epect you to find the eact area. Your grade depends on how intelligently you use ideas of high school geometry to approimate the area. As long as it makes sense, you earn full credit. The graph of looks like this: It looks as if we could use a semicircle to approimate the area beneath it it won t be a great approimation, but it will be something:

4 TEST FORM B 4 The area of the semicircle would be π ) = 9π Judging from the diagram, this is likely an underestimate. b) Use three rectangles and left endpoints to approimate A. We have = ) / =. The left endpoints are thus =, =, and =, whence A [f ) + f ) + f )] = + + ) = 4. Our approimation in part a) wasn t that bad, after all!) c) Use si rectangles and left endpoints to approimate A. We have = ) /6 = /. The left endpoints are thus =, = /, =, = /, =, and = 5 /, whence [ A f ) + f = ) + f ) + f = 4 8 = ) + f ) + f ) )] 5 d) Use the definition of the integral to find the eact value of A.

5 TEST FORM B 5 We have = ) /n and, since we use right endpoints, i = + i /n = i /n. Thus d = f i ) n = n = [ i i ) ] [ = n = = ) i n ) ] i n n ] i [ 9 i 9 n n [ 9 n n + ) 9 ] n n + ) n + ) n n n 6 [ ] 7n n + ) 7n n + ) n + ) n 6n. using L Hôpital s Rule or some other method) = = 9. e) Why do we epect c) to be more accurate than b), and d) to be eact? We epect c) to be more accurate because it uses more rectangles, and thus typically) leaves less room for error. We epect d) to be eact because, as the number of rectangles approaches, the error should correspondingly approach. 5. bonus) Suppose we know that A L is the approimation of b f ) d using n left endpoints, a and we also want to find A R, the approimation using n right endpoints. Use the geometry of these approimations to eplain how we can find A R by subtracting one easily-found value which one?) and adding one easily-found value which one?). We can find A R by removing the leftmost rectangle s area and adding the rightmost rectangle s area, because the left endpoint of one rectangle is the right endpoint of the rectangle to its immediate left. Hence A R = A L f ) }{{} leftmost rectangle + f n ) }{{} rightmost rectangle.

6 TEST FORM B 6 Left endpoints: i = a + i ) Right endpoints: i = a + i Midpoints: i = a + i ) Sum shortcuts: c = cn i = i i + ) Useful formulas i = n n + ) n + ) 6 i = n n + ) 4