Warm Up – Calculator Active 1) A particle moves along the x-axis so that at any time t > 0, its velocity is given by v(t) = 3 + 4.1cos (0.9t). What is the acceleration of the particle at time t = 4? A particle moves along the x-axis so that at any time t, 0 < t < 5, its velocity is given by When t = 0, the particle is at the origin. Write an expression for the position function, x(t), of the particle at any time t.

Riemann Sums And Trapezoidal Rule Approximating the area under a curve …or at least the area between the curve and the x-axis Riemann Sums And Trapezoidal Rule

The area under y = x2 + 1 on the interval [0,3]

Riemann Sums use rectangles to approximate Right sum – using 3 rectangles (in this case also the upper sum)

Left sum – using 3 equal subintervals (in this case also the lower sum)

Midpoint sum – 3 equal subintervals

Try finding the sum with 6 rectangles. To be more accurate… Use more rectangles! Try finding the sum with 6 rectangles.

Let f(x) = 0.5x3 – 2x2 + x + 5 Sketch the graph to illustrate the area approximation, under f(x) on the interval [-1,5] ,found by using 6 equal subintervals and an Upper Riemann Sum. b) Sketch the graph to illustrate the area approximation, under f(x) on the interval [-1,5] ,found by using 3 equal subintervals and a Right Riemann Sum.

Trapezoidal sum – 3 equal subintervals

Approximate the area under the curve y = 1- cos x on the interval using n = 4 equal subintervals and 1) Lower sums 2) Upper sums 3) Midpoint sums 4) Trapezoidal Rule 4.252 5.823 5.194 5.038

Debt(in billions of dollars) Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t Years since 1990 M Debt(in billions of dollars) 120 1 130 4 225 6 250 10 640 Use the trapezoidal rule with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

Debt(in billions of dollars) Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t Years since 1990 M Debt(in billions of dollars) 120 1 130 4 225 6 250 10 640 Use a upper Reimann sum with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

Debt(in billions of dollars) Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t Years since 1990 m Debt(in billions of dollars) 120 1 130 4 225 6 250 10 640 Use a lower Reimann sum with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate the area under the curve given by the data in the table.