Differential calculus Let C(t) be the number of people in the US that have cance

Question

Differential calculus Let C(t) be the number of people in the US that have cancer as a function of years since 1950, t. Interpret the derivative of C at 67. State the definition of dC/dt (t). Use the definition of the derivative to calculate dC/dt (t) if C(t) = 100 + t^2. (You will get no credit for using the power rule here.) Write the definition of the derivative of c(t) = 100t^2 middot e^.01t Present the product rule, power rule, chain rule, and constant multiple rule. (These are theorems. That is, these are facts.) Use those rules to calculate the derivative of c(t) = 100t^2 middot e^.01t, indicating where you use each rule.

Explanation / Answer

1)

the change in number of people in US in 1967

2)

change in number of people in the US with change in time

3)

dC/dt = lim h->0 100+(t+h)^2 - 100 -t^2 /h

= lim h->0 h^2+2th/h

= lim h->0 h + 2t

= 2t

5)

d/dx x^n = n x^n-1

d/dx (uv) = u d/dx v + v d/dx u

d/dx f(g(x)) = d/dx f(g(x)) d/dx g(x)

6)

dc/dt = 100t^2 * -0.1 * e^-0.1t + e^-0.1t 200t

=>

= t e^-0.1t ( 200 - 10t)

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