Gradient at a point Problem set A

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

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

Find the derivative at the given value.

1. (a) \(A=4+3h^{-1}\) when \(h=2\)
2. (b) \(\psi=2\theta^{-1}+\theta^3\) when \(\theta=0.5\)

Find the gradient of the tangent to the curve.

1. (a) \(y=2x^3-x\) at the point where \(x=-3\)
2. (b) \(y=4x^2+x\) at the point where \(x=0.5\)

Find the gradient of the tangent to the curve.

1. (a) \(y=x^2+\frac{5}{x}\) at the point where \(x=2\)
2. (b) \(y=3x^3-\frac{2}{x}\) at the point where \(x=0.5\)

Find the rate of change by differentiating.

1. (a) How quickly does \(p=2s^3\) change as s changes, when \(s=2\)?
2. (b) What is the rate of increase of Z with respect to k when \(k=-2\), if \(Z=-5k^2\)?

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

1. (a) The curve \(y=x^4-2x\), gradient \(30\).
2. (b) The curve \(y=2x+\frac{18}{x}\), gradient \(-16\).

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

1. (a) Find the average rate of change of f between x = 1 and x = 2.
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.
3. (c) Find \(f'(1)\). What do the answers to the two parts above approach?

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

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.

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.
2. (b) Find the speed of the rocket at t = 2.
3. (c) Explain why the two answers are different.

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

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