Homework #10 Solutions

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1 MAT Fall Homework # Solutions Problems Bolded problems are worth points. Section 5.:, 6, 8,, Section 5.:, 6,, 8,, Notes: On 5.., evaluate the integral using the fnint function (available through MATH 9 ) on a TI graphing calculator, or through another computation system. On 5.., the sums should cover the interval from t = 5 to t = A car starts moving at time t = and goes faster and faster. Its velocity is shown in the following table. Estimate how far the car travels during the seconds. t (sec) 6 9 v (ft/sec) Solution: Since the car is accelerating, its velocity function is increasing, so we can get an underestimate for the distance traveled by taking a left-hand sum over -second intervals: L = = ft. Similarly, we can get an overestimate with a right-hand sum: L = = 65 ft. A better estimate is usually obtained from averaging the left- and right-hand estimates, + 65 which in this case gives = 5.5 ft Figure 5.8 shows the velocity, v, of an object (in meters/sec). Estimate the total distance the object traveled between t = and t = 6. Solution: We estimate values for v(t) at t =,,..., 6: t 5 6 v(t) 6 8 Then the left-hand and right-hand sums for these values are L = = 5, R = = 6, and their average is (5 + 6) = m. Estimates may differ if different values of v(t) are determined.

., the sums should cover the interval from t = 5 to t =. 5... A car starts moving at time t = and goes faster and faster. Its velocity is shown in the following table.

2 MAT Fall The following table gives world oil consumption, in billions of barrels per year. Estimate total oil consumption during this 5-year period. Year Oil (bn bbl/yr) Solution: We again average the left-hand and right-hand sum estimates for these data: L = = 597., R = = 6.. Then their average is ( ) = 6.5 billion barrels Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 5 minutes. Roger starts out strong, but after an hour an a half he is so exhausted that he has to stop. Jeff s data follow: Time (min) Speed (mph) 8 7 (a) Assuming that Roger s speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half-hour. (b) Give upper and lower estimates for the distance Roger ran during the entire hour and a half. Solution (a): Since Roger is decelerating, his velocity is decreasing, so a left-hand sum will give us an overestimate (and a right-hand one, an underestimate). To make the units correct, we convert the time intervals from 5 minutes to of an hour when we compute the sum. For the first half-hour, we use only two intervals: L = + = = 5.75, R = + = = 5.5. Therefore, Roger traveled at least 5.5 miles and at most Solution (b): We take left- and right-hand sums for the entire 9-minute interval: L = ( ) =.5, R = ( ) =.5. Therefore, Roger ran somewhere between.5 and.5 miles.

) = 6.5 billion barrels. 5... Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 5 minutes.

3 MAT Fall 5... Figure 5. shows the rate of change of a fish population. Estimate the total change in the population during this -month period. Solution: We use the graph to estimate the rate of change every months: t (months) 6 8 r(t) (fish/month) 7 7 Then averaging the left- and right-hand sums, we have L = ( ) = 6, R = ( ) = 6, so the total change in population is probably about 6 fish Estimate dx using a left-hand sum with n =. x + Solution: With a =, b =, and n =, x = =. Then the left-hand sum uses the x-values,, and 8, so it is = = Use the table to estimate f (x) dx. What values of n and x did you use?. x f (x) Solution: Since the x-values in the table are spaced every units, we use X =. Since there are 5 values total, we use n = 5 = (since each sum ignores either a or b). Then the left- and right-hand sums gives estimates: L = ( ) = 6,5, R = ( ) = 7,55. Averaging them, we have the estimate (6,5 + 7,55) = 7, Use the graph provided to estimate f (x) dx. Solution: We estimate f (x) at multiples of between and : x 5 f (x)

+ + 7 + ) = 6, so the total change in population is probably about 6 fish. 5... Estimate dx using a left-hand sum with n =. x + Solution: With a =, b =, and n =, x = =.

4 MAT Fall These estimates are probably accurate to ±.. Then the left- and right-hand sums are L = ( ) = 7.5, R = ( ) = The average is ( ) = 7.5. (If the estimates of the f (x) values differ, or if the x-values where the f (x) estimates are taken change, this estimate will vary as well.) Using the figure below, find the value of 6 f (x) dx. f (x) 5 6 x Solution: We estimate the area under the figure from x = to x = 6 geometrically. Shading the region under the curve and cutting it into rectangles and squares, we have f (x) 5 6 x The two rectangles have area 5 and, the triangle on the left has area ()() =, and the one on the right has area ()() =. Hence, the total area is = 8.5, so this is the value of the integral Use a calculator to evaluate + x dx. Solution: Using the fnint function on the TI, we find that dx.. + x

f (x) 5 6 x Solution: We estimate the area under the figure from x = to x = 6 geometrically.

5 MAT Fall 5... Use the data in the table below from t = 5 to t = and the notation for Riemann sums. t f (t) 8 (a) If n =, what is t? What are t, t, t, t, t? What are f (t ), f (t ), f (t ), f (t ), f (t )? (b) Find the left and right sums using n =. (c) If n =, what is t? What are t, t, t? What are f (t ), f (t ), f (t )? (d) Find the left and right sums using n =. 5 Solution (a): For n =, t = =. Then t = 5, t = 7, t = 9, t =, and t =, so f (t ) =, f (t ) =, f (t ) = 8, f (t ) =, and f (t ) =. Solution (b): The left and right sums with n = are f (t i ) t = =, i= f (t i ) t = = 6. i= 5 Solution (c): For n =, t = f (t ) =, f (t ) = 8, and f (t ) =. =. Then t = 5, t = 9, and t =, so Solution (d): The left and right sums with n = are f (t i ) t = + 8 =, i= f (t i ) t = 8 + = 9. i= 5

(d) Find the left and right sums using n =. 5 Solution (a): For n =, t = =. Then t = 5, t = 7, t = 9, t =, and t =, so f (t ) =, f (t ) =, f (t ) = 8, f (t ) =, and f (t ) =.

t hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards.

t hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards. The area under a graph often gives useful information. Velocit-time graphs Constant velocit The sketch shows the velocit-time graph for a car that is travelling along a motorwa at a stead 7 mph. 7 The