Question

7. (**) (a) (5 marks) Using the substitution $x = \tan(t)$, show that $\int \frac{1}{1 + x^2} dx = \tan^{-1} x + C$. (b) (7 marks) Using a suitable substitution show that $\int \frac{a}{a^2 + x^2} dx = \tan^{-1} (\frac{x}{a}) + C$, where $a$ is some constant. (c) (8 marks) The number (in thousands), $N$, of tadpoles in a pond after $t$ days may be approximated by the solution to the equation $\frac{dN}{dt} = \frac{20}{\pi(100 + t^2)}$. Given that there are initially 0 tadpoles in the pond, find $N$ as a function of $t$. What happens to $N$ in the limit $t \to \infty$? What may we conclude about the long-time population of tadpoles? 2

          7. (**)
(a) (5 marks) Using the substitution $x = \tan(t)$, show that
$\int \frac{1}{1 + x^2} dx = \tan^{-1} x + C$.
(b) (7 marks) Using a suitable substitution show that
$\int \frac{a}{a^2 + x^2} dx = \tan^{-1} (\frac{x}{a}) + C$,
where $a$ is some constant.
(c) (8 marks) The number (in thousands), $N$, of tadpoles in a pond after $t$ days may be approximated
by the solution to the equation
$\frac{dN}{dt} = \frac{20}{\pi(100 + t^2)}$.
Given that there are initially 0 tadpoles in the pond, find $N$ as a function of $t$. What happens to
$N$ in the limit $t \to \infty$? What may we conclude about the long-time population of tadpoles?
2

7. (**)
(a) (5 marks) Using the substitution x = tan(t), show that
∫(1)/(1 + x^2) dx = tan^-1 x + C.
(b) (7 marks) Using a suitable substitution show that
∫(a)/(a^2 + x^2) dx = tan^-1 ((x)/(a)) + C,
where a is some constant.
(c) (8 marks) The number (in thousands), N, of tadpoles in a pond after t days may be approximated
by the solution to the equation
(dN)/(dt) = (20)/(π(100 + t^2)).
Given that there are initially 0 tadpoles in the pond, find N as a function of t. What happens to
N in the limit t →∞? What may we conclude about the long-time population of tadpoles?
2

Added by Michaela K.

Question

Please give Ace some feedback