Some calculus problems in the assignment of Calculus 1 course

1) a) show that e^x >= 1 + x for x>=0
b) deduce that e^x >= 1 + x + 1/2 x^2 for x >= 0
c) use mathematical induction to prove that for x >= 0 and any positive integer n,
e^x >= 1 + x + x^2 / 2! + ... + x^n / n!

2) Show that cubic function ( a third degree polynomial ) always has exactly 1 point of inflection. If it's graph has 3 x-intercepts x1, x2, x3, show that the x coordinate of the inflection point is (x1+x2+x3) / 3


i've been stuck with these problems for a while (luckily, other 100 problems are easier (Rofl)) plz help me, deadline is coming ~.~ thks a lot!!

Quote:

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Originally Posted by tracewalker

1) a) show that e^x >= 1 + x for x>=0
b) deduce that e^x >= 1 + x + 1/2 x^2 for x >= 0
c) use mathematical induction to prove that for x >= 0 and any positive integer n,
e^x >= 1 + x + x^2 / 2! + ... + x^n / n!

2) Show that cubic function ( a third degree polynomial ) always has exactly 1 point of inflection. If it's graph has 3 x-intercepts x1, x2, x3, show that the x coordinate of the inflection point is (x1+x2+x3) / 3


i've been stuck with these problems for a while (luckily, other 100 problems are easier (Rofl)) plz help me, deadline is coming ~.~ thks a lot!!

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Hints:

a. Define the following function: $\displaystyle f(x)=ln(x+1)-x$, "investigate" her.

b. $\displaystyle e^{x+0.5x^2}>?$

Quote:

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Originally Posted by tracewalker

1) a) show that e^x >= 1 + x for x>=0
b) deduce that e^x >= 1 + x + 1/2 x^2 for x >= 0
c) use mathematical induction to prove that for x >= 0 and any positive integer n,
e^x >= 1 + x + x^2 / 2! + ... + x^n / n!

2) Show that cubic function ( a third degree polynomial ) always has exactly 1 point of inflection. If it's graph has 3 x-intercepts x1, x2, x3, show that the x coordinate of the inflection point is (x1+x2+x3) / 3


i've been stuck with these problems for a while (luckily, other 100 problems are easier (Rofl)) plz help me, deadline is coming ~.~ thks a lot!!

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MHF policy is to not knowingly help with work that counts towards a student's final grade. It's meant to be student's own work, not the work of others.

Thread closed. (You can discuss this further with me via pm if you want).