Anti-differentiation: reversing the power rule Problem set B

Find f(x) in each case.

1. (a) \(f'(x)=x^{9}\)
2. (b) \(f'(x)=x^{-5}\)
3. (c) \(f'(x)=3\)

Find y in each case.

1. (a) \(\frac{dy}{dx}=18x^{8}\)
2. (b) \(\frac{dy}{dx}=\frac{7}{3}x^{6}\)

Find y in each case.

1. (a) \(\frac{dy}{dx}=-6x^{-4}\)
2. (b) \(\frac{dy}{dx}=\frac{5}{3}x^{-6}\)

Find y in each case.

1. (a) \(\frac{dy}{dx}=7x^{6} + 8x^{3} - 2\)
2. (b) \(\frac{dy}{dx}=\frac{2}{5}x^{4} - \frac{3}{2}x\)

Find \(\int\left(3x^{2} + 10x^{-6} + 8\right)dx\).

Find \(\int\left(\frac{9}{4}y^{-4} + \frac{6}{5}y^{-7}\right)dy\).

Rewrite as a power of x, then find f(x).

1. (a) \(f'(x)=-\frac{15}{x^{4}}\)
2. (b) \(f'(x)=\frac{18}{x^{7}}\)

Find f(x) in each case.

1. (a) \(f'(x)=12\sqrt{x}\)
2. (b) \(f'(x)=6x^{-1/2}\)

The water in a pond changes at a rate of \(\frac{dV}{dt}=9t^{2}+8t-6\) litres per hour, where V is the volume gained and t the time in hours.

1. (a) Find an expression for V in terms of t, including the constant c.
2. (b) No water has been added when \(t=0\), so \(V=0\). Write down the value of c.

The brightness B of a lamp changes with the distance r metres from it at a rate \(\frac{dB}{dr}=-\frac{15}{r^{4}}+6r\). Find an expression for B in terms of r.

Find f(x) when \(f'(x)=\frac{9}{x^{3}}+9\sqrt{x}-8x^{-1/2}\).

Priya writes: \(\int 9x^{-3}\,dx=-27x^{-4}+c\). Say what Priya did wrong and give the correct answer.