(1) Let f(x) be a continuous function defined on [a, b], it is known that there exists some c such that
\int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)
Explain the fact by using graph. Note that you don't need to prove the statement.
(2) Let f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n,
Prove that there exists θ such that
f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}. calculusintegrationfunctiontrigonometrygeometryrectanglecalculus computations