Question

10 5. (a) A student uses LEFT(500) to approximate Do they get an over approximation or an under approximation? Justify your answer using derivatives. 100 (b) A student uses TRAP(250) to approximate(x+1)3 dx. Do they get 0 an over approximation or an under approximation? Justify your answer using derivatives.

Answer 1

A student uses LEFT (500) to approximate integral^10_0 e^0x^2 dx Do they get an over approximation or an under approximation ? Justify your answer using derivatives Now , implies Given function integral^10_0 e^_x^2 dx let consider f(x) = e^-x^2 now , using derivatives f(x) = e^-x^2 f'(x) = e^-x^2 (-2x) = -2xe^-x^2 < 0: for x epsilon (0, 10) from that f is decreasing function on (0, 10 integral^10_0 f(x) dx lessthanorequalto LEFT(n) Hence , LEFT (500) is an over - approximation \r\n A student uses TRAD(250) to approximate integral^100_0 (9 + 1)^3 dx Do they get an over approximation or an under approximation > Justify you answer using derivatives now , given function integral^100_0 (x + 1)^3 dx let consider f(x) = (x + 1)^3 now , apply derivation f(x) = (x + 1)^3 f'(x) = 3(x + 1)^2 f''(x) = 3(2(x + 1) = 6(x + 1) > 0 for x epsilon (0, 100) therefore , f'(x) is increasing to (0, 100) and f is concave up on (0, 100) therefore integral^b_1 f(x) dx = TRAD(n)