Gradient at a point Problem set A Worked solutions

Find \(\frac{dy}{dx}\) at the given value.

1. (a) \(y=3x^4\) when \(x=2\)

   • \(\frac{dy}{dx}=12x^{3}\)
   • \(\text{When }x=2:\ \frac{dy}{dx}=12(2)^{3}=96\)

2. (b) \(y=x^3+5x^2\) when \(x=4\)

   • \(\frac{dy}{dx}=3x^{2} + 10x\)
   • \(\text{When }x=4:\ \frac{dy}{dx}=3(4)^{2}+10(4)=88\)

Find the derivative at the given value.

1. (a) \(A=4+3h^{-1}\) when \(h=2\)

   • \(\frac{dA}{dh}=-\frac{3}{h^{2}}\)
   • \(\text{When }h=2:\ \frac{dA}{dh}=-\frac{3}{(2)^{2}}=-0.75\)

2. (b) \(\psi=2\theta^{-1}+\theta^3\) when \(\theta=0.5\)

   • \(\frac{d\psi}{d\theta}=3\theta^{2} - \frac{2}{\theta^{2}}\)
   • \(\text{When }\theta=0.5:\ \frac{d\psi}{d\theta}=3(0.5)^{2}-\frac{2}{(0.5)^{2}}=-7.25\)

Find the gradient of the tangent to the curve.

1. (a) \(y=2x^3-x\) at the point where \(x=-3\)

   • \(\frac{dy}{dx}=6x^{2} - 1\)
   • \(\text{When }x=-3:\ \frac{dy}{dx}=6(-3)^{2}-1=53\)

2. (b) \(y=4x^2+x\) at the point where \(x=0.5\)

   • \(\frac{dy}{dx}=8x + 1\)
   • \(\text{When }x=0.5:\ \frac{dy}{dx}=8(0.5)+1=5\)

Find the gradient of the tangent to the curve.

1. (a) \(y=x^2+\frac{5}{x}\) at the point where \(x=2\)

   • \(\frac{dy}{dx}=2x - \frac{5}{x^{2}}\)
   • \(\text{When }x=2:\ \frac{dy}{dx}=2(2)-\frac{5}{(2)^{2}}=2.75\)

2. (b) \(y=3x^3-\frac{2}{x}\) at the point where \(x=0.5\)

   • \(\frac{dy}{dx}=9x^{2} + \frac{2}{x^{2}}\)
   • \(\text{When }x=0.5:\ \frac{dy}{dx}=9(0.5)^{2}+\frac{2}{(0.5)^{2}}=10.25\)

Find the rate of change by differentiating.

1. (a) How quickly does \(p=2s^3\) change as s changes, when \(s=2\)?

   • Rate of change = \(\frac{dp}{ds}\)
   • \(\frac{dp}{ds}=6s^{2}\)
   • \(\text{When }s=2:\ \frac{dp}{ds}=6(2)^{2}=24\)

2. (b) What is the rate of increase of Z with respect to k when \(k=-2\), if \(Z=-5k^2\)?

   • Rate of change = \(\frac{dZ}{dk}\)
   • \(\frac{dZ}{dk}=-10k\)
   • \(\text{When }k=-2:\ \frac{dZ}{dk}=-10(-2)=20\)

Find the values of x at which the curve has the given gradient.

1. (a) The curve \(y=x^4-2x\), gradient \(30\).

   • \(\frac{dy}{dx}=4x^{3} - 2\)
   • \(4x^{3} - 2=30\)
   • Solve for x: \(x=2\)

2. (b) The curve \(y=2x+\frac{18}{x}\), gradient \(-16\).

   • \(\frac{dy}{dx}=2 - \frac{18}{x^{2}}\)
   • \(2 - \frac{18}{x^{2}}=-16\)
   • Solve for x: \(x=-1\) or \(x=1\)

Let \(f(x)=x^2+5x\).

1. (a) Find the average rate of change of f between x = 1 and x = 2.

   • \(f(1)=6,\ f(2)=14\)
   • \(\frac{14-6}{2-1}=8\)

2. (b) Find the average rate of change of f between x = 1 and x = 1.1, and between x = 1 and x = 1.01.

   • \(f(1.1)=6.71\), so \(\frac{6.71-6}{0.1}=7.1\)
   • \(f(1.01)=6.0701\), so \(\frac{6.0701-6}{0.01}=7.01\)

3. (c) Find \(f'(1)\). What do the answers to the two parts above approach?

   • \(f'(x)=2x+5\), so \(f'(1)=7\)
   • The averages 8, 7.1, 7.01 approach 7, the gradient at x = 1

The depth D cm of water in a pond t hours after the tap is opened is modelled by \(D=60+12t-t^2\), for \(0\leq t\leq6\).

1. (a) How quickly is the depth changing when \(t=5\)? Give the units.

   • \(\frac{dD}{dt}=12-2t\)
   • \(t=5:\ \frac{dD}{dt}=12-2(5)=2\)
   • D is in cm and t in hours, so the rate is in cm per hour.

2. (b) At what time is the depth increasing at 6 cm per hour?

   • \(12-2t=6\)
   • \(2t=6\Rightarrow t=3\)

The cost per item, C dollars, of a print run of n items is modelled by \(C=4n+\frac{100}{n}\), for \(n>0\). Find the value of n at which C is increasing at a rate of 3 dollars per item.

• \(C=4n+100n^{-1}\Rightarrow\frac{dC}{dn}=4-\frac{100}{n^2}\)
• \(4-\frac{100}{n^2}=3\Rightarrow n^2=100\)
• \(n=\pm10\)
• n must be positive, so \(n=10\).

The height of a model rocket, h metres, t seconds after launch is modelled by \(h=5t^2+2t\) for the first 5 seconds.

1. (a) Find the average speed of the rocket between t = 2 and t = 3.

   • \(h(2)=24,\ h(3)=51\)
   • \(\frac{51-24}{3-2}=27\) m/s

2. (b) Find the speed of the rocket at t = 2.

   • \(\frac{dh}{dt}=10t+2\)
   • At t = 2: \(10(2)+2=22\) m/s

3. (c) Explain why the two answers are different.

   • The speed is increasing with t
   • So the average over 2 to 3 is more than the speed at the start, 2

Find the values of x at which the tangent to the curve \(y=x^3-6x^2+5x\) has gradient 5.

• \(\frac{dy}{dx}=3x^{2} - 12x + 5\)
• \(3x^2-12x+5=5\Rightarrow3x^2-12x=0\)
• \(3x(x-4)=0\Rightarrow x=0\text{ or }x=4\)

The curves \(y=x^3+x\) and \(y=6x^2-8x\) have the same gradient at two values of x. Find them.

• \(\frac{d}{dx}(x^3+x)=3x^2+1\)
• \(\frac{d}{dx}(6x^2-8x)=12x-8\)
• \(3x^2+1=12x-8\Rightarrow x^2-4x+3=0\)
• \((x-1)(x-3)=0\Rightarrow x=1\text{ or }x=3\)