practice another\nlet ( i=int_{0}^{x} f(x) d x ), where ( f ) is the function whose graph is shown…

practice another\nlet ( i=int_{0}^{x} f(x) d x ), where ( f ) is the function whose graph is shown below.\n(a) use the graph to find ( l_{2}, r_{2} ) and ( m_{2} ).\n( l_{2}= )\n( r_{2}= )\n( m_{2}= )\n(b) are these underestimates or overestimates of ( i ) ?\n( \bigcirc l_{2} ) is an underestimate.\n( \bigcirc l_{2} ) is an overestimate.\n( \bigcirc r_{2} ) is an underestimate.\n( \bigcirc r_{2} ) is an overestimate.\n( \bigcirc m_{2} ) is an underestimate.\n( \bigcirc m_{2} ) is an overestimate.\n(c) use the graph to find ( t_{2} ).\n( t_{2}= )\nhow does it compare with ( i ) ?\n( t_{2} ) ? ( i )\n(d) for any value of ( n ), list the numbers ( l_{n}, r_{n}, m_{n}, t_{n} ) and ( i ) in increasing order. (enter your answers as a comma-separated list. enter your answer using the variables rather than numerical values.)

practice another\nlet ( i=int_{0}^{x} f(x) d x ), where ( f ) is the function whose graph is shown below.\n(a) use the graph to find ( l_{2}, r_{2} ) and ( m_{2} ).\n( l_{2}= )\n( r_{2}= )\n( m_{2}= )\n(b) are these underestimates or overestimates of ( i ) ?\n( \bigcirc l_{2} ) is an underestimate.\n( \bigcirc l_{2} ) is an overestimate.\n( \bigcirc r_{2} ) is an underestimate.\n( \bigcirc r_{2} ) is an overestimate.\n( \bigcirc m_{2} ) is an underestimate.\n( \bigcirc m_{2} ) is an overestimate.\n(c) use the graph to find ( t_{2} ).\n( t_{2}= )\nhow does it compare with ( i ) ?\n( t_{2} ) ? ( i )\n(d) for any value of ( n ), list the numbers ( l_{n}, r_{n}, m_{n}, t_{n} ) and ( i ) in increasing order. (enter your answers as a comma-separated list. enter your answer using the variables rather than numerical values.)

Answer

Explanation:

Step1: Recall the formulas for (L_{n},R_{n},M_{n},T_{n})

  • Left - Riemann sum: (L_{n}=\Delta x\sum_{i = 0}^{n - 1}f(x_{i})), where (\Delta x=\frac{b - a}{n}), (a = 0), (b = 4), (n = 2), so (\Delta x=\frac{4-0}{2}=2), (x_{0}=0), (x_{1}=2). (L_{2}=2(f(0)+f(2))). From the graph, (f(0)=0.5), (f(2)=2), so (L_{2}=2(0.5 + 2)=5).

  • Right - Riemann sum: (R_{n}=\Delta x\sum_{i = 1}^{n}f(x_{i})), (x_{1}=2), (x_{2}=4). (R_{2}=2(f(2)+f(4))). From the graph, (f(2)=2), (f(4)=3.2), so (R_{2}=2(2 + 3.2)=10.4).

  • Mid - point Riemann sum: (M_{n}=\Delta x\sum_{i = 1}^{n}f(\frac{x_{i-1}+x_{i}}{2})), (x_{0}=0), (x_{1}=2), (x_{2}=4). The mid - points are (x_{0.5}=1), (x_{1.5}=3). (M_{2}=2(f(1)+f(3))). From the graph, (f(1)=1.2), (f(3)=2.8), so (M_{2}=2(1.2+2.8)=8).

  • Trapezoidal rule: (T_{n}=\frac{\Delta x}{2}(f(x_{0})+2f(x_{1})+f(x_{2}))), (\Delta x = 2), (x_{0}=0), (x_{1}=2), (x_{2}=4). (T_{2}=\frac{2}{2}(f(0)+2f(2)+f(4))=(0.5 + 4+3.2)=7.7).

  • Under - estimate/Over - estimate: Since (y = f(x)) is concave up.

    • For (L_{n}): The function is increasing. (L_{n}) uses the left - hand side of the sub - intervals. Since the function is increasing, (L_{2}) is an underestimate.
    • For (R_{n}): Since the function is increasing, (R_{2}) is an overestimate.
    • For (M_{n}): For a concave - up function, (M_{2}) is an underestimate.
    • For (T_{n}): For a concave - up function, (T_{2}) is an overestimate.

  • Ordering: For a concave - up and increasing function, the general order is (L_{n}<M_{n}<I<T_{n}<R_{n})

Answer:

(a) (L_{2}=5), (R_{2}=10.4), (M_{2}=8) (b) (L_{2}) is an underestimate, (R_{2}) is an overestimate, (M_{2}) is an underestimate (c) (T_{2}=7.7), (T_{2}>I) (d) (L_{n},M_{n},I,T_{n},R_{n})