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The gross domestic product​ (in billions of​ dollars) can be approximated by ​p(t)equals564 plus t left parenthesis 36 t superscript 0.6 baseline minus 103 right parenthesis​, where t is the number of years since 1960. ​a) find upper p prime left parenthesis t right parenthesis. ​b) find upper p prime left parenthesis 45 right parenthesis. ​c) in​ words, explain what upper p prime left parenthesis 45 right parenthesis represents.

Respuesta :

We have been given gross domestic product​ (in billions of​ dollars) can be approximated by [tex]P(t)=564+t(36t^{0.6}-101)[/tex].

(a) In this part, we need to compute the derivative of this function:

[tex]P'(t)=\frac{d}{dt}(564+t(36t^{0.6}-103))\\P'(t)=\frac{d}{dt}(564)+\frac{d}{dt}(t(36t^{0.6}-103))\\P'(t)=0+57.6t^{0.6}-103\\[/tex]

[tex]P'(t)=57.6t^{0.6}-103[/tex]

(b) In this part, we need to find the value of P'(45). So, we will substitute t=45

[tex]P'(45)=57.6(45)^{0.6}-103\\P'(45)=565.3924-103\\P'(45)=462.39[/tex] Billion dollars per year.

(c) P'(45)=462.39 represents that 45 years after 1960, that is, in 2005, the GCP was changing at a rate of 462.39 billion dollars per year.