Barrons AP Calculus

(D) This    is  theorem (2).    Prove   by  counterexamples that    (A),    (B),    (C),    and

(D) are false.

(A) This    is  a   restatement of  the Fundamental Theorem.    In  theorem (1),

interchange t   and x.

(D) Apply   theorem (1),    noting  that

.

(E) Let  ;  then

By  the Chain   Rule,    ,  where   theorem (1) is  used    to

find     .  Replace u   by  x^2.

(E) Since   dx  =   −4  sin θ   dθ, you get the new integral    .

Use theorem (4) to  get the correct answer.

(E) L(5)    =   9   ·   5   +   11  ·   7   +   7   ·   8   +   3   ·   4   +   1   ·   6.

(D) The average value   is  .