How to find instantaneous rate of change for #y=4x^3+2x-3# at x=2?

Answer 1

#50#

We need to differentiate the expression, #4x^3+2x-3# to find the slope of the tangent, #d/dx[4x^3+2x-3]# = #12x^2+2#, [ by the general power rule for differentiation, i.e, if #y=ax^n#, #dy/dx=anx^[n-1]#] and so when #x=2#, #dy/dx# =#12[2]^2+2# = #50#.
This is the rate of change of #y# with respect to #x# at the point where #x=2#, and means #y# is changing fifty times faster than #x# at this point. Hope this was helpful.
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Answer 2

#50#

"Instantaneous rate of change" is just a fancy way of saying "derivative". We need to differentiate this business and plug in #2# at the end.

We can find the derivative using the power rule. Here, we multiply the constant times the exponent, and the power gets decremented. Doing this, we get

#y'=12x^2+2#
NOTE: Recall that the derivative of an #x# term is just its coefficient, and the derivative of a constant is #0#.
Now, we can plug #2# in for #x# to get
#=12(2)^2+2#
#=48+2#
#=50#
The instantaneous rate of change at #x=2# is #50#.

Hope this helps!

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Answer 3

To find the instantaneous rate of change of the function ( y = 4x^3 + 2x - 3 ) at ( x = 2 ), you need to find the derivative of the function ( y ) with respect to ( x ), which gives you the slope of the tangent line at ( x = 2 ). Then, substitute ( x = 2 ) into the derivative to find the instantaneous rate of change at that point.

Taking the derivative of ( y ) with respect to ( x ) gives:

( \frac{dy}{dx} = 12x^2 + 2 )

Now, substitute ( x = 2 ) into the derivative:

( \frac{dy}{dx} \bigg|_{x=2} = 12(2)^2 + 2 = 48 + 2 = 50 )

So, the instantaneous rate of change of ( y = 4x^3 + 2x - 3 ) at ( x = 2 ) is ( 50 ).

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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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