Calculus Answers

Let f(x)={3x^2, x lessthan or equal to1;ax+b, x graterthan 1
Find the value of a and b so that f is differentiable at x=1.

a)Let f(x)=tan x
i)Show that there exists no c in the interval ]0, pi[such that f'(c)=0 even though f(0)=f(pi)=0
ii)Explain why the result in part (a) does not violate Rolle's Theorem.

Let s(t)=t^3+6t^2 be the position function of a particle moving along an axis, where s is in metres and t is in seconds. Find the the instantaneous acceleration a(t), and show the graph of acceleration with time.

Find the area of the surface that is generated by revolving the portion of the curve y=x^2 between x=1 and x=2 about the y-axis.

Trace the curve x = a sin 2t (1+cos2t),
y=acos2t(1-cos2t), 0<=t<=π

For what value of a and b will
f(x)={█((x^2-4)/(x-2); x≤2@〖ax〗^2-bx+3; 2≤x≤3@2x-a+b; x≥3)┤
Be continuous at R

Find the following limits, if they exist.
lim┬(x→1)⁡〖(x^4-1)/(x-1)〗
lim┬(x→0)⁡〖(2-cos⁡3x-cos⁡4x)/x〗
lim┬(x→8)⁡〖(x-8)/(∛x-2)〗
lim┬(r→0)⁡〖r^2/(1-cos^2⁡r )〗
lim┬(x→1)⁡〖(x|x-1|)/(x-1)〗
lim┬(x→0)⁡〖(1-cos⁡4θ)/(1-cos⁡6θ )〗
lim┬(t→0)⁡〖(√(2+t)-√2)/t〗
lim┬(x→0)⁡〖sin⁡〖(sin⁡x)〗/x〗
lim┬(x→1.5)⁡〖(〖2x〗^2-3x)/|2x-3| 〗
lim┬(θ→0)⁡〖sin⁡θ/(θ-tan⁡θ )〗
lim┬(x→2)⁡〖(x-2)/(x^2-8)〗
lim┬(x→1)⁡cos⁡〖((x^2-1)/(x-1))〗

Check whether the limit of the function 6 2
3
2
3
( , )
x y
x y
f x y
+
= exists as )0,0(

A company is required to fence off the area around a robot arm to comply with health and safety law. They have 750m of fencing available. Find the maximum area they can fence off?