5.2 Further Differentiation

Find  for each of the following:

Differentiate with respect to , simplifying your answers as far as possible:

sin

A curve has the equation 

Find.

Hence find the gradient of the normal to the curve at the point , giving your answer correct to 3 decimal places.

Consider the curve with equation 

Find 

Hence find the equation of the tangent to the curve at the point (−2,1), giving your answer in the form , where and are integers.

Let  where  and 

Find the equation of the tangent of  at 

A curve has the equation 

Find expressions for  and 

Determine the coordinates of the local minimum of the curve.

The diagram below shows part of the graph of  where  is the function defined by

Points  and are the three places where the graph intercepts the  -axis.

Find 

Show that the coordinates of point are

Find the equation of the tangent to the curve at point 

Let 

Find 

Find 

Determine the ranges of -values for which the graph of  is

giving all boundary values for the ranges as exact values.

Hence find the exact of the points of inflection for the graph of . Be sure to show that any points identified are indeed points of inflection.

Let  where 

Find the number of points containing a horizontal tangent.

Show algebraically that the gradient of the tangent at  is 

State the gradient of the tangent at 

It can be found that as the function,  undergoes a transformation  the number of stationary points found between   increases.

Find the number of stationary points on  after a transformation of and hence, state the general rule representing the number of stationary points in terms of  where

Let  and  for 

Solve 

Use the quotient rule to show that the derivative of  is  

Consider the function defined by 

Find 

Show that

Using your answers to parts (b) and (c), determine the -coordinates of any

on the curve .

An international mission has landed a rover on the planet Mars. After landing, the rover deploys a small drone on the surface of the planet, then rolls away to a distance of 6 metres in order to observe the drone as it lifts off into the air. Once the rover has finished moving away, the drone ascends vertically into the air at a constant speed of 2 metres per second.

Let  be the distance, in metres, between the rover and the drone at time  seconds. 

Let  be the height, in metres, of the drone above the ground at time seconds. The entire area where the rover and drone are situated may be assumed to be perfectly horizontal.

Find

the height of the drone above the ground at the moment when the distance between the rover and the drone is increasing at a rate of  .

Find an expression for the derivative of each of the following functions:

Consider the function  defined by . 

Show that  is decreasing everywhere on its domain.

Consider the function  defined by . 

Point A is the point on the graph of  for which the -coordinate is   .

Show that the equation of the tangent to the graph of at point A may be expressed in the form 

Point B is the point on the graph of  at which the normal to the graph is vertical.

Show that the coordinates of the point of intersection between the tangent to the graph of at point A and the tangent to the graph of  at point B are

Consider the function  defined by 

Show that the normal line to the graph of  at   intercepts the -axis at the point

Let  ,  where and  are real-valued functions such that

for all .

Given that   and ,  where  find the distance between the  -intercept of the tangent to the graph of  at  and the -intercept of the normal to the graph of  at .  Give your answer in terms of  and/or  as appropriate.

Consider the curve with equation  defined for all  , where    is a positive integer.

For the case where , find the number of points in the interval  at which the curve has a horizontal tangent.

In terms of , state in general how many (i) turning points and (ii) points of inflection the curve will have in the interval .  Give a reason for your answers.

Let ,  where  and  are well-defined functions with anywhere on their common domain. 

By first writing  ,  use the product and chain rules to show that

Consider the function  defined by , ,  where  is a positive integer.

Show that the graph of will have no points of inflection in the case where .

Show that, for , the second derivative of is given by

Hence show that the graph of will only have points of inflection in the case where is an odd integer greater than or equal to 3.  In that case, give the exact coordinates of the points of inflection, giving your answer in terms of   where appropriate.  In your work you may use without proof the fact that for odd integers with 

A small conical flask, in the shape of a right cone stood on its flat base, is being filled with perfume via a small hole at its vertex.  The cone has a height of 6 cm and a radius of 2 cm. 

Perfume is being poured into the flask at a constant rate of 0.3 cm³s^-1.

Find the rate of change of the depth of the perfume in the flask at the instant when the flask is half full by volume.

A large block of ice is being prepared for use by a team of ice sculptors.  The block is in the shape of a cuboid with the ratio of its length to width to height being equal to  1 : 2 : 5. The block melts uniformly such that its surface area decreases at a constant rate, losing  of surface area every hour.  You may assume that as the block melts, its shape remains a cuboid with the dimensions in the same ratio to each other as in the original cuboid.

The block of ice is considered stable enough to be sculpted so long as the loss of volume due to melting does not exceed a rate  per hour. 

Find, in terms of , the volume of the largest block of ice that can be used for ice sculpting under such conditions.