course Mth 173
If function f ' (x) is the derivative of the function f(x), then the function f(x) is an antiderivative of the function f ' (x). Hint: What is the derivative of x^3?
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3x^2
What is the derivative of 5 x^3?
5(3x^2)
15x^2
What is the derivative of x^3 / 12?
3x^2/12
If the derivative of an unknown function is x^2, then what is the function?
That function would be an antiderivative of x^2.
x^3/3
What would be an antiderivative of .0723 x^3?
.0723(x^4)/4
Submit a copy of this document and include your best answers to these questions. If it then becomes clear how to find an antiderivative of .001t^2 + .14t + 1.8, go ahead and give me your reasoning on this question also.
.001t^2 + .14t + 1.8
.001(t^3)/3 + .14(t^2)/2 + 1.8
But would the denominator be 3 + 2? So the antiderivative would be .001(t^3) + .14(t^2) + 1.8 / 3 + 2??"
No. .001 t^3 + .14 t^2 + 1.8/3 + 2 would have derivative .003 t^2 + .28 t, not .001t^2 + .14t + 1.8.
Your expression .001(t^3)/3 + .14(t^2)/2 + 1.8 was very nearly correct, but the derivative of this expression would be .001 t^2 + .14 t.
To get 1.8 in your derivative, your antiderivative function would also require the term 1.8 t.
A correct antiderivative is .001(t^3)/3 + .14(t^2)/2 + 1.8t.
Any function .001(t^3)/3 + .14(t^2)/2 + 1.8t + c, where c is a constant number, would also be an antiderivative (note that the derivative of a constant is 0--a constant number does not change).