I need help for solving these qusitions please 1-The slope of the tangent line t

Question

I need help for solving these qusitions please

1-The slope of the tangent line to the graph of the exponential function y=10x at the point (0,1) is limh%u2192010h%u22121h.
Estimate the slope to three decimal places.

2-The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table below. Find lower and upper estimates of the distance she covered during the first three seconds of the race. Use left- and right- rectangles. Do not use midpoint rectangles. In order to get the best estimates, use 0.5 rectangle widths.

Enter your answer as a Riemann sum to show your work.

The lower estimate is
The upper estimate is

3-a. Estimate %u222B606%u2212x2dx using 3 left rectangles.   

b. Estimate %u222B606%u2212x2dx using 3 right rectangles.   

c. Estimate %u222B606%u2212x2dx using 3 midpoint rectangles.   

4-For the function shown below, estimate %u222B60f(x)dx using

a. 3 left rectangles   

b. 3 right rectangles   

c. 3 midpoint rectangles   

5-For the function f described by the following table, estimate

%u222B600f(x)dx using the following. Enter each answer as a sum to show your work.

5 left rectangles   

5 right rectangles   

5 midpoint rectangles   

Explanation / Answer

I'm answering 4th and 5th question as all others are already answered in the question itself. Here are the answers/solutions. 4.a) integral using 3 left rectangles. Since 3 rectangles are to be used and the integral is to be calculated from x=0 to x=6. So, width of each rectangle would be 6/3 = 2. The 3 rectangles would be from (x=0 to x=2),(x=2 to x=4),(x=4 to x=6). Now, since left rectangles are to be used, so the height of rectangles would be determined by the value of f(x) at their respective left boundary, i.e., 1, 4 and 1 for the 3 rectangles respectively.(Calculated from the graph). Now, area of these 3 rectangles would be (1*2 = 2),(4*2 = 8) and (1*2 = 2) respectively. The sum of these 3 areas gives the estimated value of the integral over f(x) from x = 0 to x = 6. This would be equal to 2 + 8 + 2 = 12 4.b) integral estimation using 3 right rectangles. In this case, the height of the rectangles would be determined by their respective right boundaries, i.e., the values of f(x) at the right boundaries of these rectangles. These would be 4(f(2)), 1(f(4)) and 5(f(6)) respectively. So, total area = 2*4 + 2*1 + 2* 5 = 20 4.c) integral estimation using 3 mid-point rectangles. In this case the height of the rectangles would be determined by their mid points respectively,i.e., value of f(x) at the mid-points of these rectangles. These would be 3(f(1)), 3(f(3)) and 3(f(5)) respectively. So total area = 2*3 +2*3 +2*3 = 18 5.a) integral using 5 left rectangles. Since 5 rectangles are to be used and the integral is to be calculated from x=0 to x=60. So, width of each rectangle would be 60/5 = 12. The 5 rectangles would be from (x=0 to x=12),(x=12 to x=24),(x=24 to x=36),(x=36 to x=48) and (x=48 to x=60). Now, since left rectangles are to be used, so the height of rectangles would be determined by the value of f(x) at their respective left boundary, i.e., f(0),f(12), f(24),f(36),f(48) respectively.(to be Calculated from the table). Now, total area of these 5 rectangles would be =(12*f(0) + 12*f(12) + 12*f(24) + 12*f(36) + 12*f(48) = 12*5 + 12*40 + 12*9 + 12*9 + 12*31 = 12(5 + 40 + 9 + 9 +31) = 12 * 94 = 1128 The sum of these 5 areas gives the estimated value of the integral over f(x) from x = 0 to x = 60. This would be equal to 1128 5.b) integral estimation using 5 right rectangles. In this case, the height of the rectangles would be determined by their respective right boundaries, i.e., the values of f(x) at the right boundaries of these rectangles. These would be 40(f(12)), 9(f(24)), 9(f(36)), 31(f(48)) and 30(f(60)) respectively. So, total area = 12*40 + 12*9 +1 2* 9 + 12*31 + 12*30 = 12(40 + 9 + 9 + 31 + 30) = 12 * 119 = 1428 4.c) integral estimation using 5 mid-point rectangles. In this case the height of the rectangles would be determined by their mid points respectively,i.e., value of f(x) at the mid-points of these rectangles. These would be 13(f(6)), 28(f(18)), 19(f(30)), 40(f(42) and 2(f(44)) respectively. So total area = 12*13 +12*28 +12*19 + 12*40 + 12*2 = 12(13 + 28 +19 +40 +2) = 12 * 102 = 1224

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.