Expanding, splitting and increasing or decreasing Problem set A One question at a time

Differentiate with respect to x. The other letters are constants.

1. (a) \(y=8ax-\frac{5b}{x^2}\)

   • \(y=8ax - \frac{5b}{x^{2}}\)
   • \(\frac{dy}{dx}=8a + \frac{10b}{x^{3}}\)
   • Letters other than x are constants.

2. (b) \(y=4b^2x^3+\frac{6a}{cx}\)

   • \(y=4b^{2}x^{3} + \frac{6a}{cx}\)
   • \(\frac{dy}{dx}=12b^{2}x^{2} - \frac{6a}{cx^{2}}\)
   • Letters other than x are constants.

Expand, then differentiate with respect to x.

1. (a) \(y=(bx)^2\)

   • \(y=(bx)^2=b^{2}x^{2}\)
   • \(\frac{dy}{dx}=2b^{2}x\)

2. (b) \(y=(4ax)^2\)

   • \(y=(4ax)^2=16a^{2}x^{2}\)
   • \(\frac{dy}{dx}=32a^{2}x\)

Expand, then find \(\frac{dy}{dx}\). The other letters are constants.

1. (a) \(y=(x+p)(x+q)\)

   • \(y=(x+p)(x+q)=x^{2} + (p+q)x + pq\)
   • \(\frac{dy}{dx}=2x + p + q\)

2. (b) \(y=(px+q)(qx+p)\)

   • \(y=(px+q)(qx+p)=pqx^{2} + (p^{2}+q^{2})x + pq\)
   • \(\frac{dy}{dx}=2pqx + p^{2} + q^{2}\)

Split each fraction into powers of x, then differentiate with respect to x.

1. (a) \(y=\frac{x+12}{x}\)

   • \(y=\frac{x+12}{x}=1 + \frac{12}{x}\)
   • \(\frac{dy}{dx}=-\frac{12}{x^{2}}\)

2. (b) \(y=\frac{3-x}{x^2}\)

   • \(y=\frac{3-x}{x^2}=-\frac{1}{x} + \frac{3}{x^{2}}\)
   • \(\frac{dy}{dx}=\frac{1}{x^{2}} - \frac{6}{x^{3}}\)

3. (c) \(y=\frac{x^2+4x^3}{2x}\)

   • \(y=\frac{x^2+4x^3}{2x}=2x^{2} + \frac{1}{2}x\)
   • \(\frac{dy}{dx}=4x + \frac{1}{2}\)

Find the values of x for which the function is decreasing.

1. (a) \(f(x)=2x^{2} - 8x + 5\)

   • \(f'(x)=4x - 8\)
   • \(4x - 8<0\)
   • \(4x - 8=0\Rightarrow x=2\)
   • The sign of the x term decides the side: \(x<2\)

2. (b) \(f(x)=-x^{2} + 10x - 3\)

   • \(f'(x)=-2x + 10\)
   • \(-2x + 10<0\)
   • \(-2x + 10=0\Rightarrow x=5\)
   • The sign of the x term decides the side: \(x>5\)

Find the values of x for which the function is increasing.

1. (a) \(f(x)=x^{2} - 6x\)

   • \(f'(x)=2x - 6\)
   • \(2x - 6>0\Rightarrow x>3\)

2. (b) \(f(x)=x^2+bx+c\), where \(b\) and \(c\) are constants

   • \(f'(x)=2x+b\)
   • \(2x+b>0\Rightarrow x>-\frac{b}{2}\)
   • The constant c differentiates to 0, so it does not appear in the answer.

Differentiate \(s=8t-\frac{9}{t}\) with respect to \(t\).

• \(s=8t - \frac{9}{t}\)
• \(\frac{ds}{dt}=8 + \frac{9}{t^{2}}\)

If \(3x+y=5x^2-2\), find \(\frac{dy}{dx}\).

• \(y=5x^{2} - 3x - 2\)
• \(\frac{dy}{dx}=10x - 3\)

The heat energy H joules needed to warm a metal bar by T °C is modelled by \(H=cmT\), where c and m are constants.

1. (a) Find the rate at which H increases with T.

   • \(H=cmT\)
   • \(\frac{dH}{dT}=cm\)
   • c and m are constants, so \(cmT\) differentiates like \(5T\).

2. (b) Find this rate when \(c=0.9\) and \(m=2.5\). Give the units.

   • \(0.9\times2.5=2.25\)
   • H is in joules and T in °C, so the rate is in joules per °C.

The temperature \(\theta\) °C of a wall at distance d metres from a fire is modelled by \(\theta=\frac{k}{d}\), where k is a constant. The cooling rate is the rate of change of \(\theta\) with respect to d.

1. (a) Find an expression for the cooling rate in terms of k and d.

   • \(\theta=kd^{-1}\)
   • \(\frac{d\theta}{dd}=-kd^{-2}=-\frac{k}{d^2}\)

2. (b) Find an expression for the cooling rate in terms of \(\theta\) and k.

   • \(d=\frac{k}{\theta}\)
   • \(-\frac{k}{(k/\theta)^2}=-\frac{\theta^2}{k}\)

3. (c) Point P is halfway between the fire and a point Q. Find the value of (cooling rate at P) ÷ (cooling rate at Q).

   • \(\text{At Q: }-\frac{k}{d^2}.\ \text{At P (distance }\tfrac12d):\ -\frac{k}{(d/2)^2}=-\frac{4k}{d^2}\)
   • \(\text{ratio}=4\)

A model for the height H cm of a plant after d days is \(H=5+qd+2qd^2\), where q is a constant and \(d\geq0\).

1. (a) Find the rate of change of H with respect to d.

   • \(H=5+qd+2qd^2\)
   • \(\frac{dH}{dd}=q+4qd=q(1+4d)\)

2. (b) Explain why you would expect H to be an increasing function. What does this tell you about q?

   • The height of a growing plant does not fall, so the gradient should be positive: \(\frac{dH}{dd}>0\).
   • \(q(1+4d)>0\)
   • For \(d\geq0\), \(1+4d\) is positive, so q must be positive.

Show that \(f(x)=3x^3+4x-7\) is increasing for all values of x.

• \(f'(x)=9x^{2} + 4\)
• \(x^2\geq0\Rightarrow 9x^2+4\geq4>0\)
• f′(x) is positive for all x, so f is increasing for all x.

The constant k is positive. Let \(f(x)=\frac{x^3+kx}{2x^2}\).

1. (a) Find \(f'(x)\).

   • \(f(x)=\frac{x^3}{2x^2}+\frac{kx}{2x^2}=\frac{1}{2}x+\frac{k}{2}x^{-1}\)
   • \(f'(x)=\frac12-\frac{k}{2}x^{-2}=\frac12-\frac{k}{2x^2}\)

2. (b) Given that \(f'(2)=\frac14\), find the value of k.

   • \(f'(2)=\frac12-\frac{k}{8}=\frac14\)
   • \(\frac{k}{8}=\frac14\Rightarrow k=2\)