Gradient at a point Problem set B

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

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

Find the derivative at the given value.

1. (a) \(K=7-2n^{-1}\) when \(n=4\)
2. (b) \(\omega=\phi^4-\phi^{-2}\) when \(\phi=0.5\)

Find the gradient of the tangent to the curve.

1. (a) \(y=x^4-3x\) at the point where \(x=-2\)
2. (b) \(y=5x^2-2x\) at the point where \(x=1.5\)

Find the gradient of the tangent to the curve.

1. (a) \(y=x^2+\frac{8}{x}\) at the point where \(x=4\)
2. (b) \(y=2x^3-\frac{5}{x}\) at the point where \(x=0.5\)

Find the rate of change by differentiating.

1. (a) How quickly does \(h=3r^4\) change as r changes, when \(r=1\)?
2. (b) What is the rate of increase of Z with respect to k when \(k=-3\), if \(Z=-7k^2\)?

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

1. (a) The curve \(y=x^4+3x\), gradient \(-29\).
2. (b) The curve \(y=5x+\frac{8}{x}\), gradient \(3\).

Let \(g(x)=x^2+7x\).

1. (a) Find the average rate of change of g between x = 2 and x = 3.
2. (b) Find the average rate of change of g between x = 2 and x = 2.1, and between x = 2 and x = 2.01.
3. (c) Find \(g'(2)\). What do the answers to the two parts above approach?

The temperature \(\theta\) °C of the water in a kettle t minutes after the kettle is switched on is modelled by \(\theta=15+18t-t^2\), for \(0\leq t\leq8\).

1. (a) How quickly is the temperature changing when \(t=4\)? Give the units.
2. (b) At what time is the temperature increasing at 4 °C per minute?

A boat uses F litres of fuel per hour when it travels at v km/h, where \(F=5v+\frac{72}{v}\), for \(v>0\). Find the speed at which F is increasing at a rate of 3 litres per hour for each extra km/h.

The money M euros in a savings jar, d days after the jar was started, is modelled by \(M=0.5d^2+4d\) for the first 30 days.

1. (a) Find the average rate at which the jar fills between d = 10 and d = 12.
2. (b) Find the rate at which the jar is filling at d = 10.
3. (c) Explain why the two answers are different.

Find the values of x at which the tangent to the curve \(y=x^3-9x^2+2x\) has gradient 2.

The curves \(y=2x^3-x\) and \(y=9x^2-13x\) have the same gradient at two values of x. Find them.