How do you find the instantaneous rate of change of #g(t)=3t^2+6# at t=4?

Answer 1

Compute the first derivative and evaluate it at #t = 4#

#g'(4) = 24#

Determine the initial derivative:

#g'(t) = 6t#
Evaluate it at #t = 4#
#g'(4) = 24#
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Answer 2

It depends on what you have in your mathematical toolbox.

If you have learned the power rule, constant multiple rule and derivative of a constant, you can quickly find the derivative of #g#.
#g'(t) = 3(2x^1)+0 = 6t#.
To find the instantaneous rate of change at a particular value of #t#, evaluate the derivative at that value of #t#.
At #t=4# the instantaneous rate of change is #g'(4) = 6(4) = 24#.

If you are using a definition then it depends on the particular definition you are using.

There are several ways to express the definition.

One way of expressing it is to give:

The rate of change of #g# with respect to #t# at #t=4# is
#lim_(trarr4)(g(t)-g(4))/(t-4)#.

Another is

The rate of change of #g# with respect to #t# at #t=4# is
#lim_(hrarr0)(g(4+h)-g(4))/h#.

Still another is

The rate of change of #g# with respect to #t# at #t# is
#lim_(hrarr0)(g(t+h)-g(t))/h#.
(After we find this, we evaluate at #t=4#.

Here is the work for the first definition above.

#lim_(trarr4)(g(t)-g(4))/(t-4) =lim_(trarr4) ([3t^2+6]-[3(4)^2+6])/(t-4)# (Observe that is we substitute #t = 4#, we get the indeterminate form #0/0#.)
# = lim_(trarr4) (3t^2+6-48-6)/(t-4)# #" "# (Still #0/0#)
# = lim_(trarr4) (3t^2-48)/(t-4)#
# = lim_(trarr4) (3(t^2-16))/(t-4)#
# = lim_(trarr4) (3(t+4)(t-4)))/(t-4)#
# = lim_(trarr4) 3(t+4)#
# = 3(4+4) = 24#
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Answer 3

To find the instantaneous rate of change of ( g(t) = 3t^2 + 6 ) at ( t = 4 ), you can calculate the derivative of ( g(t) ) with respect to ( t ) using the power rule, which states that the derivative of ( t^n ) with respect to ( t ) is ( nt^{n-1} ). Applying this rule to ( g(t) ), we get:

[ g'(t) = \frac{d}{dt}(3t^2 + 6) = 6t ]

Next, substitute ( t = 4 ) into the derivative equation:

[ g'(4) = 6(4) = 24 ]

So, the instantaneous rate of change of ( g(t) = 3t^2 + 6 ) at ( t = 4 ) is ( 24 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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