derivative?

Question

What is the derivative of t/1+t^2 and then how do I know what its local max and local mins are?

Answer 1

derivitive is

1 + 2t

maximum value is when this value equals 1. Therefore, the max value occurs when t = 0.

The minimum value is when this value equals 0. Therefore, the min value occurs when t = -1/2.

Answer 2

y = t + t^2
y' = 1 + 2t --- In both cases use the power rule, multiply the power by the coefficient and then subtract one from the power

To find maxes and mins, you use the first derivative test. To do this, find when the first derivative is either undefined or equal to zero. This will give you intervals.

y' = 0 when t = -1/2

Now you need to see if the first derivative is negative or positive in the intervals (-∞, -1/2) (-1/2, ∞), so:

In the first interval y' is negative, and in the second interval it is positive. This lets you know that there is a min. when t= -1/2 because a neg. first derivative means a decreasing function, and a pos. first derivative measn an increasing function. Thus, at t= -1/2 the original function changes from decreasing to increasing, giving a minimum.

Answer 3

Derivative is (1-t^2)/(1+t^2)^2

Answer 4

Use

(f/g)' = (f'g - g'f)/g^2

The derivative will be

(1(1+t^2) - (2t)t)/(1+t^2)^2

= (1 - t^2)/(1+t^2)^2

It is 0 at t = +- 1. So that is where the maximum /minimum will be.

for t>1, the function obviously decreases as t increases, so t = +1 is a maximum.

for t < -1, its magnitude decreases with increase in magnitude of t, >> it increases as t decreases, so t = -1 is a minimum.

(Since it is antisymmetric about (0,0), if it is maximum at +1, it will be minimum at -1)

Answer 5

f(t) = t + t^2

f'(t) = 1 + 2t

the rule of thumb for DERIVATIVES is to take your exponent (in this case....(t) would have an exponent of 1) and multiply it by your coefficient. then subract your exponent by 1 and you have your new exponent.

so...

t ==> (1)(t^1) ---- 1 is my coefficient, 1 is my exponent

take down my exponent and multiply by my coefficient

(1)(1)(t^1) ==> (1)(t^1)

next, take my exponent, and subtract one.

since 1 is my exponent, 1-1 = 0

(1)(t^0), and since t^0 = 1

(1)(1) = 1
============================

(1)(t^2), 1 is my coefficient, 2 is my exponent

mulitply exponent by coefficient

(2)(1)(t^2) = (2)(t^2)

subtract one from exponent

2-1 = 1, so

(2)(t^1)
====================