Calculus 1 Weiqi Zhou
a particle moves toward a fixed direction along a line, its position at time is
then its speed is
from time to time ，the distance it traveled is , which is also
i.e.,
if is continuous on ，then it has an anti-derivative on
if is an anti-derivative of , then
where the RHS can also be written as or
if is continuous on ，then let
by MVT for definite integrals，for any , there is with
consequently
let ，find
using MVT: , s.t.
using the chain rule: let
，find
using the product rule
exercise：Find using MVT
, find
find
if is continuous and positive，show that is monotonically increasing for
take
then ,
any anti-derivative of can be written as , where is a constant
if is determined by ，find
if is continuous on ，and ，show that there is only one root for in
if is continuous on ， is differentiable, is a bijection, ，then
let
if , and is continuous on , then
show that
if are continuously differentiable on ，then
take ，, then