True or False? The chain rule is used when differentiating composite functions.

True. The chain rule is used when differentiating composite functions.

When is the product rule used?

The product rule is used when differentiating the product of two functions .

True or False? The quotient rule is used when differentiating a fraction where only the numerator is a function of x .

False. The quotient rule is used when differentiating a fraction where both the numerator and denominator are functions of x .

Define the term rate of change .

A rate of change is a measure of how a quantity is changing with respect to another quantity.

What is meant by related rates of change ?

Related rates of change are rates of change connected by a linking variable or parameter , often time. For example, if a container is being filled with water the rates of change for the volume and height of the water in the container are related rates of change connected by time.

True or False? A positive rate of change always indicates a decrease .

False. A positive rate of change indicates an increase .

What is the second order derivative of a function?

The second order derivative of a function is the derivative of the derivative of the function. It is often just called the second derivative of the function.

How can the term second derivative be defined in terms of rates of change?

The second derivative is the rate of change of the rate of change of a function. It is the rate of change of the gradient of the graph of the function.

State two uses of second order derivatives.

Two uses of second order derivatives are: testing for local minimum and maximum points, determining the concavity of a function.

True or False? The second derivative can be used to determine the nature of stationary points .

True. The second derivative can be used to determine the nature of stationary points .

True or False? The second derivative is always continuous if the first derivative is continuous.

False. The second derivative may be discontinuous even if the first derivative is continuous.

True or False? Local minimum and maximum points are types of stationary points .

True. Local minimum and maximum points are types of stationary points .

What is a local minimum point ?

A local minimum point is a point where the function value is the lowest in the immediate vicinity .

True or False? A local maximum point is the point at which a function takes on its maximum value.

False. A local maximum point is not necessarily the point at which a function takes on its maximum value . A local maximum point is a point at which the function value is the highest in the immediate vicinity . There may however be other points (not in the immediate vicinity) at which the function takes on a higher value.

How can you use the first derivative to determine if a stationary point is a local minimum or local maximum ?

To determine if a stationary point is a local minimum or local maximum , look at the gradient of the function (i.e. the value of the first derivative ) on either side of the stationary point. If the gradient is positive to the left of the point and negative to the right , then the stationary point is a local maximum . If the gradient is negative to the left of the point and positive to the right , then the stationary point is a local minimum .

What is a turning point ?

A turning point is a type of stationary point where the function changes from increasing to decreasing, or vice versa.

True or False? There is no difference between stationary points and turning points .

False. There is a difference between stationary points and turning points . A turning point is always a stationary point, but a stationary point is not always a turning point (e.g., a point of inflection can be a stationary point, but it is not a turning point).

Concavity is the way in which a curve (or surface) bends.

True or False? A curve can be both concave up and concave down in different intervals.

True. A curve can be both concave up and concave down in different intervals.

IBMathsDPApplications & Interpretation (AI)HLFlashcards5. CalculusFurther Differentiation

Further Differentiation (DP IB Applications & Interpretation (AI))

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What is the derivative of $f (x) = \sin x$ ?

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What is the derivative of $f (x) = \sin x$ ?
The derivative of $f (x) = \sin x$ is $f^{'} (x) = \cos x$ .
This is in your exam formula booklet.
What is the derivative of $f (x) = \cos x$ ?
The derivative of $f (x) = \cos x$ is $f^{'} (x) = - \sin x$ .
This is in your exam formula booklet.
What is the derivative of $f (x) = \tan x$ ?
The derivative of $f (x) = \tan x$ is $f' (x) = \frac{1}{\cos^{2} x}$ .
This is in your exam formula booklet.
What is the derivative of $f (x) = e^{x}$ ?
The derivative of $f (x) = e^{x}$ is $f^{'} (x) = e^{x}$ .
This is in your exam formula booklet.
What is the derivative of $f (x) = \ln x$ ?
The derivative of $f (x) = \ln x$ is $f^{'} (x) = \frac{1}{x}$ .
This is in your exam formula booklet.
True or False?
The derivative of $y = \sin (a x + b)$ is $\frac{d y}{d x} = a \cos (a x)$ .
False.
The derivative of $y = \sin (a x + b)$ is $\frac{d y}{d x} = a \cos (a x + b)$ .
This is not in your exam formula booklet.
What is the derivative of $y = \cos (a x + b)$ ?
The derivative of $y = \cos (a x + b)$ is $\frac{d y}{d x} = - a \sin (a x + b)$ .
This is not in your exam formula booklet.
What is the derivative of $y = \tan (a x + b)$ ?
The derivative of $y = \tan (a x + b)$ is $\frac{d y}{d x} = \frac{a}{\cos^{2} (a x + b)}$ .
This is not in your exam formula booklet.
What is the derivative of $y = e^{a x + b}$ ?
The derivative of $y = e^{a x + b}$ is $\frac{d y}{d x} = a e^{a x + b}$ .
This is not in your exam formula booklet.
What is the derivative of $y = \ln (a x + b)$ ?
The derivative of $y = \ln (a x + b)$ is $\frac{d y}{d x} = \frac{a}{a x + b}$ .
This is not in your exam formula booklet.
True or False?
The derivative of $y = \ln (a x)$ is $\frac{d y}{d x} = \frac{a}{x}$ .
False.
The derivative of $y = \ln (a x)$ is $\frac{d y}{d x} = \frac{1}{x}$ .
The derivative of $\ln (a x)$ is the same as the derivative of $\ln x$ .
True or False?
The derivative of $y = \sin (f (x))$ is $\frac{d y}{d x} = f^{'} (x) \cos (f (x))$ .
True.
The derivative of $y = \sin (f (x))$ is $\frac{d y}{d x} = f^{'} (x) \cos (f (x))$ .
This is not in your exam formula booklet.
What is the derivative of $y = \cos (f (x))$ ?
The derivative of $y = \cos (f (x))$ is $\frac{d y}{d x} = {- f}^{'} (x) \sin (f (x))$ .
This is not in your exam formula booklet.
What is the derivative of $y = \tan (f (x))$ ?
The derivative of $y = \tan (f (x))$ is $\frac{d y}{d x} = \frac{f^{'} (x)}{\cos^{2} (f (x))}$ .
This is not in your exam formula booklet.
What is the derivative of $y = e^{f (x)}$ ?
The derivative of $y = e^{f (x)}$ is $\frac{d y}{d x} = f' (x) e^{f (x)}$ .
This is not in your exam formula booklet.
What is the derivative of $y = \ln (f (x))$ ?
The derivative of $y = \ln (f (x))$ is $\frac{d y}{d x} = \frac{f^{'} (x)}{f (x)}$ .
This is not in your exam formula booklet.
What is the chain rule?
The chain rule states that if $y$ is a function of $u$ , and $u$ is a function of $x$ , then $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ .
This formula is in the exam formula booklet.
What is the product rule?
The product rule states that if $y = u v$ is the product of two functions $u$ and $v$ , where $u$ and $v$ are both functions of $x$ , then $\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$ .
This formula is in the exam formula booklet.
What is the quotient rule?
The quotient rule states that if $y = \frac{u}{v}$ is the quotient of two functions $u$ and $v$ , where $u$ and $v$ are both functions of $x$ , then $\frac{d y}{d x} = \frac{(v \frac{d u}{d x} - u \frac{d v}{d x})}{v^{2}}$ .
This formula is in the exam formula booklet.
True or False?
The chain rule is used when differentiating composite functions.
True.
The chain rule is used when differentiating composite functions.
When is the product rule used?
The product rule is used when differentiating the product of two functions.
True or False?
The quotient rule is used when differentiating a fraction where only the numerator is a function of x.
False.
The quotient rule is used when differentiating a fraction where both the numerator and denominator are functions of x.
True or False?
The chain rule can be applied multiple times in trickier problems.
True.
The chain rule can be applied multiple times in trickier problems.
E.g. when differentiating a 'function within a function within a function' like $e^{\cos (3 x - 4)}$ .
True or False?
Which function you call ' $u$ ' and which function you call ' $v$ ' is not important when using the quotient rule.
False.
Which function you call ' $u$ ' and which function you call ' $v$ ' is important when using the quotient rule.
- $u$ must refer to the numerator of $y = \frac{u}{v}$
- $v$ must refer to the denominator of $y = \frac{u}{v}$
This is due to the minus sign in the numerator of the quotient rule formula, as well as the $v^{2}$ in the denominator.
True or False?
Which function you call ' $u$ ' and which function you call ' $v$ ' is not important when using the product rule.
True.
Which function you call ' $u$ ' and which function you call ' $v$ ' is not important when using the product rule.
When using the chain rule to differentiate the function $\sin (x^{2} - 7)$ , what should you call ' $y$ ' and what should you call ' $u$ '?
When using the chain rule to differentiate the function $\sin (x^{2} - 7)$ , you should let $y = \sin u$ , and let $u = x^{2} - 7$ .
Then $\frac{d y}{d u} = \cos u = \cos (x^{2} - 7)$ , and $\frac{d u}{d x} = 2 x$ .
Those can be put into the chain rule formula $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ to find the derivative.
Define the term rate of change.
A rate of change is a measure of how a quantity is changing with respect to another quantity.
What is meant by related rates of change?
Related rates of change are rates of change connected by a linking variable or parameter, often time.
For example, if a container is being filled with water the rates of change for the volume and height of the water in the container are related rates of change connected by time.
True or False?
A positive rate of change always indicates a decrease.
False.
A positive rate of change indicates an increase.
How is the chain rule used in solving related rates of change problems?
In solving related rates of change problems, the chain rule is used to set up an equation linking rates of change.
E.g. $\frac{d V}{d t}$ and $\frac{d r}{d t}$ can be linked using the chain rule equation $\frac{d V}{d t} = \frac{d V}{d r} \times \frac{d r}{d t}$ .
State the chain rule equation for related rates of change, connecting $\frac{d y}{d x}$ , $\frac{d y}{d t}$ and $\frac{d x}{d t}$ .
The chain rule equation for related rates of change, connecting $\frac{d y}{d x}$ , $\frac{d y}{d t}$ and $\frac{d x}{d t}$ is $\frac{d y}{d t} = \frac{d y}{d x} \times \frac{d x}{d t}$ .
This is not in the exam formula booklet (but it is a variant of the general chain rule formula which is in the booklet).
What is a useful mathematical relationship between $\frac{d y}{d x}$ and $\frac{d x}{d y}$ ?
A useful mathematical relationship between $\frac{d y}{d x}$ and $\frac{d x}{d y}$ is $\frac{d y}{d x} = \frac{1}{(\frac{d x}{d y})}$ .
This allows $\frac{d y}{d x}$ to be found by finding $\frac{d x}{d y}$ instead and then inverting it.
What is the second order derivative of a function?
The second order derivative of a function is the derivative of the derivative of the function.
It is often just called the second derivative of the function.
How can the term second derivative be defined in terms of rates of change?
The second derivative is the rate of change of the rate of change of a function. It is the rate of change of the gradient of the graph of the function.
What is the notation for the second order derivative?
The notation for the second order derivative is $\frac{d^{2} y}{d x^{2}}$ or $f^{''} (x)$ .
State two uses of second order derivatives.
Two uses of second order derivatives are:
1. testing for local minimum and maximum points,
2. determining the concavity of a function.
True or False?
The second derivative can be used to determine the nature of stationary points.
True.
The second derivative can be used to determine the nature of stationary points.
True or False?
The second derivative is always continuous if the first derivative is continuous.
False.
The second derivative may be discontinuous even if the first derivative is continuous.
What is a stationary point?
A stationary point is a point on a function where $f^{'} (x) = 0$ .
I.e., it is a point on a function where the tangent is horizontal.
True or False?
Local minimum and maximum points are types of stationary points.
True.
Local minimum and maximum points are types of stationary points.
What is a local minimum point?
A local minimum point is a point where the function value is the lowest in the immediate vicinity.
True or False?
A local maximum point is the point at which a function takes on its maximum value.
False.
A local maximum point is not necessarily the point at which a function takes on its maximum value.
A local maximum point is a point at which the function value is the highest in the immediate vicinity. There may however be other points (not in the immediate vicinity) at which the function takes on a higher value.
How can you use the first derivative to determine if a stationary point is a local minimum or local maximum?
To determine if a stationary point is a local minimum or local maximum, look at the gradient of the function (i.e. the value of the first derivative) on either side of the stationary point.
- If the gradient is positive to the left of the point and negative to the right, then the stationary point is a local maximum.
- If the gradient is negative to the left of the point and positive to the right, then the stationary point is a local minimum.
What is a turning point?
A turning point is a type of stationary point where the function changes from increasing to decreasing, or vice versa.
True or False?
There is no difference between stationary points and turning points.
False.
There is a difference between stationary points and turning points.
A turning point is always a stationary point, but a stationary point is not always a turning point (e.g., a point of inflection can be a stationary point, but it is not a turning point).
How do you find the x-coordinates of stationary points?
To find the x-coordinates of any stationary points a function $f (x)$ might have, solve the equation $f^{'} (x) = 0$ .
True or False?
The second derivative test can always determine the nature of a stationary point.
False.
The second derivative test can not always determine the nature of a stationary point.
The second derivative test fails when $f^{''} (x) = 0$ at the stationary point.
What does $f^{''} (x) < 0$ at a stationary point indicate?
If $f^{''} (x) < 0$ at a stationary point, it means that the stationary point is a local maximum point.
What does $f^{''} (x) > 0$ at a stationary point indicate?
If $f^{''} (x) > 0$ at a stationary point, it means that the stationary point is a local minimum point.
When should you use the first derivative test for stationary points?
You should use the first derivative test when the second derivative test is inconclusive (i.e., when $f^{''} (x) = 0$ at the stationary point).
True or False?
A global minimum point represents the lowest value of $f (x)$ for all values of $x$ .
True.
A global minimum point represents the lowest value of $f (x)$ for all values of $x$ .
What is concavity?
Concavity is the way in which a curve (or surface) bends.
What does concave down mean?
A curve is concave down if $f^{''} (x) < 0$ for all values of $x$ in an interval.
What does concave up mean?
A curve is concave up if $f^{''} (x) > 0$ for all values of $x$ in an interval.
True or False?
A curve can be both concave up and concave down in different intervals.
True.
A curve can be both concave up and concave down in different intervals.
What is a point of inflection?
A point of inflection is a point at which the graph of $y = f (x)$ changes concavity.
True or False?
All points of inflection are stationary points.
False.
Not all points of inflection are stationary points.
Some points of inflection have $f^{'} (x) \neq 0$ .
True or False?
If $f^{''} (x) = 0$ , then $x$ is always a point of inflection.
False.
$f^{''} (x) = 0$ is necessary for a point to be a point of inflection. However it is possible for a point to have $f^{''} (x) = 0$ but not be a point of inflection.
How can you find the x-coordinates of possible points of inflection?
The x-coordinates of possible points of inflection of a function $f (x)$ are the solutions of the equation $f^{''} (x) = 0$ .
In addition to $f^{''} (x) = 0$ , what additional step is needed to confirm a point of inflection?
To confirm a point of inflection, in addition to $f^{''} (x) = 0$ you need to check that $f^{''} (x)$ changes sign on either side of the point.

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