Calculus I help?

Question

-Suppose f(t)=4t^2-t+3 is a distance function. What expression represents the a.average b.instantaneous velocity?
-Explain why there is acceleration for a ball moving at a constant speed on a circular track.
-What is the derivative of sin^2(x)?
-What is the derivative of √x?

Answer 1

1. a. The average velocity over a given period of time is the distance traveled, divided by the time elapsed. So for the average velocity between time 1 (t1) and time 2 (t2), you would calculate:
([4(t2)^2-(t2)+3] - [4(t1)^2-(t1)+3]) / (t2 - t1)

Notice that if t1 and t2 are very small (e.g., 1/16 and 1/8), the average velocity will be negative. Apparently the distance function is decreasing for small values of t. (Read on for an explanation.)

1.b. The instantaneous velocity is the rate of change of f(t) with respect to t. That is, it is f'(t), the derivative of f(t), which is: 8t - 1. Note that this is negative when t is less than 1/8 (as predicted in the preceding paragraph).

2. In order to be moving at a constant velocity, an object has to be travelling at a constant speed in a STRAIGHT LINE. Any change from a constant speed in a straight line involves acceleration. In this case, the acceleration is toward the center of the circular track. The ball is constantly having its velocity changed (accelerated) in the direction of the center of the track in order to keep the ball from continuing in a straight line (i.e., "going off on a tangent").

3. To take this derivative, use the chain rule:
derivative of sin^2(x) =
2 sin(x) times derivative of sin(x) =
2 sin(x) cos(x) =
sin(2x)
[Note: Some other responders have given you the wrong answer to this part of your question.]

4. √x = x^(1/2)
Derivative of x^(1/2) = (1/2) x^(-1/2) = (1/2) x^(1/2) / x =
(1/2) √x / x

Answer 2

1) avg vel. = Δd / Δt = [ f (t final) - f (t initial)] / [(t final) - (t initial)]
instantaneous velocity = f ' (t) = d/dt (4t^2 - t + 3) =
8t - 1

2) It has centripetal acceleration. Acceleration can be linear ( change in speed) or rotational (change in direction). Since it is going in a circle, it is constantly turning, and constantly accelerating.

3) 2 sin x cos x (chain rule), hopefully with respect to x, be careful with the notation

4) 1 / (2√x), power rule, d/dx x^1/2 = (1/2) x^(1/2 - 1)

Answer 3

Average velocity = distance / time
Average velocity = (4t² - t + 3) / t
Instantaneous velocity = f `(t) = 8t - 1

Could this be radial acceleration?

f(x) = (sin x)²
f ` (x) = 2.sin x.cos x = sin 2x

f (x) = x^(1/2)
f `(x) = (1/2).x^(-1/2) = 1 / (2x^(1/2))

Answer 4

distance/time=vellocity
d4t^2-t+3/dt=vellocity[inst.......is d vellocity at dat time,when t=0]
2x
-cos^2x

Answer 5

y= √x = x^1/2
y'= 1/2x^(-1/2)

Answer 6

solve your homeworks by yourself, yahoo answers wasn't estaplished for such reasons.