5.1 Differentiation

Consider the function

Find

Find the gradient of the graph of at 

Find the coordinates of the points at which the normal to the graph of has a gradient of 4.

The equation of a curve is .

Find the equation of the tangent to the curve at 

Give your answer in the form .

Find the coordinates of the points on the curve where the gradient is .

Consider the function .

Calculate

A line,  is tangent to the graph of  at the point  .

Find the equation of . Give your answer in the form .

The graph of  and  have a second intersection at point .

Use your graphic display calculator to find the coordinates of .

Consider the function .

Find .

The equation of the tangent line to the graph  at  is .

Calculate the value of .

Calculate the value of and write down the function.

The curve with equation  has a gradient of 7 at the point  and a gradient of 3 at the point

By considering show that and

Hence find the values of  and .

By considering a point that you know to be on the curve, find the value of .

The curve  has equation  The point  lies on .

Find an expression for

Show that an equation of the normal to at point  is

This normal cuts the -axis at the point .

Find the length of , giving your answer as an exact value.

Find the values of for which is an increasing function.

Show that the function is increasing for all

The graph of the cubic function  is shown below. Point  a local minimum, is located at the origin and point , a local maximum, sits at the point .

State the equations of the horizontal tangent to the curve.

Write down the value of  where the point of inflection is located.

Find the intervals where is decreasing.

Sketch the graph of  labelling clearly any intercepts and axis of symmetry.

The diagram below shows part of the curve with equation . The curve touches the -axis at  and cuts the -axis at . The points  and  are stationary points on the curve.

Using calculus, and showing all your working, find the coordinates of and .

Show that  is a point on the curve and explain why those must be the coordinates of point

The equation of the curve  is . A section of the curve  is shown on the diagram below.

Find .

There are two points, and , along the curve  at which the gradient of the normal to the curve  is equal to .

Calculate the -coordinates of points  and .

Find the -coordinates of the stationary points on the graph with equation 

Find the nature of the stationary points found in part (a).

Determine the -coordinate of the point of inflection on the graph with equation

Explain why, in this case, the point of inflection is not a stationary point.

The graph of a continuous function has the following properties:

The function is concave down in the interval 

The function is concave up in the interval

The graph of the function intercepts the -axis at the points and  where and  are such that

The coordinates of the turning points of the function are  and , which are such that

The graph of the function intercepts the -axis at 

Given that the value of the function is positive when , sketch a graph of the function. Be sure to label the -axis with the -coordinates of the stationary points and the point of inflection, and also to label the points where the graph crosses the coordinate axes.

An engineer is designing a right cone that is to be produced on a 3D printer. The cone has a base radius of  cm and a height of  cm, and while the radius may vary freely the height must always be 7 cm more than the radius.

Write down, in terms of  only, the formula for the volume of the cone.

Find the exact value of the radius at the point where the instantaneous rate of change of the volume with respect to the radius is . 

A curve has the equation

Points  and  are the two points on the curve where the gradient is equal to 3, and the   -coordinate of  is less than zero.

Find the coordinates of points and .

Find the equations of

(i)     the tangent to the curve at point .

(ii)    the normal to the curve at point .

Point  is the point of intersection of the two lines found in part (b).

Find the coordinates of point . Give your answers as exact fractions.

A curve has equation  .

The gradient of the tangent to the curve at the point  is 25.

The gradient of the tangent to the curve at the point  is .

Find the values of , ,  and .

The diagram below shows a part of the graph of the function  ,  where

Calculate the average rate of change of  between and

(i)     

(ii)   

(iii)   

Explain what would happen to the values of the average rates of change in part (b) if you continued to calculate them, moving the second  value closer and closer to 3 each time.

Let be a function defined by for all

The curve intercepts the -axis at points and  where 

Find the values of  and

The curve has a local minimum at point .

Show that the -coordinate of point  is equal to Be sure to justify that the point corresponding to that -coordinate is indeed a local minimum, and that it is the only local minimum.

The curve  has a point of inflection at point

Find the gradient of the normal to the curve at point

A function  is defined by

for all real numbers .

Show that the curve has a stationary point when  and determine the corresponding -coordinate.

Find the -coordinates of any other stationary points on the curve.

Determine the ranges of values of  for which  is

Given that the value of the function when is greater than the value of the function at any other stationary point, sketch the curve of  Be sure to show clearly the -coordinates of any minimum and maximum points, as well as the coordinates of the point where the curve crosses the -axis.

A function  is defined for all for The derivative of  is given by

The graph of is concave up when ,  where  is the least possible number that makes that inequality true.

Find the value of .

Show that the curve  has only one point of inflection, and find the gradient of the normal line to the curve at that point.

A curve has equation  where  and

Show that the curve will only have stationary points if and satisfy the inequality

In the case where the inequality in part (a)(i) is satisfied, determine the -coordinate(s) of the stationary point(s), giving your answer as simply as possible in terms of and

Show that the curve will always have exactly one point of inflection, and determine its -coordinate in terms of  and b.

In the case where the point of inflection is also a stationary point, show that the curve will have no other stationary points.

In the case where the curve has two distinct stationary points, show that the -coordinate of the point of inflection will lie halfway between the -coordinates of the two stationary points.