Calculus!!!! HELP!!!?

Question

Given f(x) = {-x, for x> and /or = 1
{x+k, for x< 1


a). find the value of k so that f(x) will be continuous at x=1
b). Using the value of k found in part A, is f(x) differentiable at x=1


AND!
What is the minimus initial velocity needed so that a ball thrown upwards from the ground reaches the top of a building 40 feet high?


AND!!
A cylindrical Terrarium hanging from the ceiling leaks sand at the rate of 5cm^3/minute. The sand falls to the floor, forming a conical pile. the radius and the height of the cone are in the ratio 3:2. How fast is the height of the pile increasing when the radius is 9cm??


OMG....im SO lost! i dont even know where to start on any of these problems..PLEASE help me!

Answer 1

Hello,

The terrarium :

This cylinder has a volume denoted by;

v = pi*r^2*h. but 3/2 =r/h then h = 2r/3 thus v = 2/3 *pi*r^3

So dv/dt = 2*pi*r^2 dr/dt


Second one.

S(t) = -16t^2 + vi*t + c where t = time of flight, vi is the initial velocity and d is the distance above ground from which it was thrown. Now

s(t) = -16t^2 + v1*t + 0 since it was thrown from the ground.
Now s(t) = -16t^2 + v1t then the distance s(t) = 40 so we have
40 = -16T^2 + v1t then 0 = -16t^2 + v1t - 40

Now this is a parabola opening upward so we can derivative or -b/(2a) to find the maximum. then ds/dt = -32t + v1 then setting this = 0 give us v1 = 32t so s(t) = -16t^2 +32t then when s(t) = 40 we have 40 = -16t^2 +32 t^2 or 40 = 16t^2 then
t^2 = 40/16 = 5/2 so t = +/- (1/2)*sqrt10 of course we throw out the negative value so t = .5sqrt10

Now Put this in for t in 40 = -16t^2 + v1t and we have

40 = -16*((5/2) + v1*(.5sqrt10)
I'll let you finish the math.




now 5 = 2*pi*9^2 dr/dt

5/508.87 = dr/dt

Then since h = 2r/3 dh/dt = 2/3 dr/dt I'll let you put in the values from here.

a) For the function to be continuous at x = 1 the function must have value there and the limit from both sides must be the same. So f(x) = -x at x = 1 is -1. Then the limit as x ---> 1 from the right +1 must be -1 so x + k = -1 then as x ---> 1 we have 1 + k = -1 or k = -2.




Hope This Helps!!

Answer 2

i might want to assert structure is area of engineering which signifies that a sturdy carry close on math is needed, pre-cal is is like algebra to the subsequent element with the Pi chart and understanding about radians and perspective measures (which looks substantial) yet Calculus is the learn of derivatives, (the slope of a line at a particular element) type of unnecessary no be counted what container of workd your going to, in case you question me.