Differentiation (Edexcel IGCSE Maths A)

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  • Define the term gradient.

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Cards in this collection (43)

  • Define the term gradient.

    Gradient is a measure of steepness, referring to how steep a line or curve is on a graph. It describes the rate at which y changes with respect to x.

  • True or False?

    The gradient of a curve is always constant.

    False.

    The gradient of a straight line is constant.

    The gradient of a curve changes as the value of x changes.

  • Define the term tangent in the context of a curve on a graph.

    In the context of a curve on a graph, a tangent is a straight line that touches the curve at one point (without otherwise cutting across the curve near that point).

  • What is the gradient function?

    The gradient function, also known as the derivative or derived function, is an algebraic function that takes inputs of x-coordinates and gives outputs of gradients at the points with those x-coordinates.

  • What does fraction numerator straight d y over denominator straight d x end fraction represent?

    fraction numerator straight d y over denominator straight d x end fraction represents the gradient function or derivative. It is pronounced "dy by dx" or "dy over dx".

  • Define differentiation.

    Differentiation is the operation that turns curve equations into gradient functions.

  • What is the main rule for differentiating terms with powers of x?

    To differentiate powers of x:

    • multiply the number in front by the power

    • then subtract one from the power

    So the derivative of 2 x cubed is  open parentheses 3 cross times 2 close parentheses x to the power of 3 minus 1 end exponent equals 6 x squared.

  • What is the rule for differentiating a term like 10x?

    When differentiating a term like 10x (where there is no power written next to the x), remember that this means 10x1, so will differentiate to 1×10x0, which is 10.

  • True or False?

    Any constant term (number on its own) differentiates to zero.

    True.

    Any constant term (number on its own) differentiates to zero.

  • What is the main rule for differentiating a number of terms added or subtracted together? (For example something like 2 x cubed minus 3 x plus 4.)

    The main rule for differentiating a number of terms added or subtracted together is simply to differentiate each term one at a time.

    So the derivative of  2 x cubed minus 3 x plus 4  is  open parentheses 3 cross times 2 close parentheses x to the power of 3 minus 1 end exponent minus 3 plus 0 equals 6 x squared minus 3.

  • What are the steps to find the gradient of a curve at a particular point using the gradient function?

    To find the gradient of a curve at a particular point using the gradient function:

    1. Find the x-coordinate of the point on the curve

    2. Use differentiation to find the gradient function for the curve

    3. Substitute the x-coordinate into the gradient function

  • What is the definition of a stationary point?

    A stationary point is a point at which the gradient of a curve is zero.

  • What is the definition of a turning point?

    A turning point is a point at which a curve changes from moving upwards to moving downwards, or vice versa.

    I.e. it is a maximum (peak) or minimum (trough) on the curve.

  • What condition does the gradient of a curve satisfy at a turning point?

    At a turning point, the gradient of a curve is equal to zero.

  • State an equation involving the derivative that can be used for finding the x-coordinate of a turning point.

    At a turning point the derivative is equal to zero, so an equation that can be used to find the x-coordinate of a turning point is  fraction numerator straight d y over denominator straight d x end fraction equals 0.

    (To find the x-coordinate(s) of the turning point(s), solve that equation for x.)

  • True or false?

    The term 'stationary point' is also used for turning points.

    True.

    Turning points are also referred to as stationary points. A stationary point is any point on a curve where the gradient is equal to zero.

  • What are the steps to find the coordinates of a turning point?

    To find the coordinates of a turning point:

    1. Set the derivative (gradient function) equal to zero and solve to find the x-coordinate

    2. Substitute the x-coordinate into the equation of the curve to find the y-coordinate

    (In step 2, be sure to use the original equation of the curve, not the gradient function!)

  • True or false?

    After using a gradient function to find the x-coordinate of a turning point, you should substitute that x-coordinate back into the gradient function to find the y-coordinate of the turning point.

    False.

    After using a gradient function to find the x-coordinate of a turning point, you should substitute that x-coordinate back into the original equation of the curve to find the y-coordinate of the turning point.

  • Define a maximum point.

    A maximum point is a type of stationary point (or turning point) where a graph reaches the top of a "peak".

    (It is sometimes called a local maximum point, because there may be other parts of the graph that reach higher values.)

  • Define a minimum point.

    A minimum point is a type of stationary point (or turning point) where the graph reaches the bottom of a "trough".

    (It is sometimes called a local minimum point, because there may be other parts of the graph that reach lower values.)

  • True or False?

    A positive parabola (positive x squared term) always has exactly one maximum point.

    False.

    A positive parabola always has exactly one minimum point (and no maximum points).

  • True or False?

    A negative parabola (negative x squared term) always has exactly one maximum point.

    True.

    A negative parabola always has exactly one maximum point (and no minimum points).

  • True or False?

    A positive cubic curve (positive x cubed term) has a maximum point on the left and a minimum point on the right.

    True.

    A positive cubic curve (positive x cubed term) has a maximum point on the left and a minimum point on the right.

  • True or False?

    A negative cubic curve (negative x cubed term) has a maximum point on the left and a minimum point on the right.

    False.

    A negative cubic curve (negative x cubed term) has a minimum point on the left and a maximum point on the right.

  • How can you use the gradient function (derivative) to classify a turning point?

    At a maximum point:

    • the gradient just before the turning point is positive

    • the gradient at the turning point is zero

    • the gradient just after the turning point is negative

    At a minimum point:

    • the gradient just before the turning point is negative

    • the gradient at the turning point is zero

    • the gradient just after the turning point is positive

  • True or False?

    You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.

    True.

    You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.

  • Without doing any calculations, how do you know that the graph of y equals x squared minus 10 x plus 26 has a minimum point and not a maximum point?

    The graph of y equals x squared minus 10 x plus 26 has a positive x squared term so it has the shape of a positive parabola (quadratic curve).

    This means it is a U shaped curve, so must have a minimum point (not a maximum point).

  • The area of a shape, A m2, is given by A equals x squared minus 6 x plus 14.

    Explain how you would use differentiation to find the minimum area.

    The area of a shape, A m2, is given by A equals x squared minus 6 x plus 14.

    To find the minimum area

    1. Differentiate A to get fraction numerator straight d A over denominator straight d x end fraction

    2. Set fraction numerator straight d A over denominator straight d x end fraction equals 0 to form and solve an equation in x

    3. Substitute this value of x back into A

    4. This must be the minimum value of A as A equals x squared minus 6 x plus 14 is a positive quadratic curve (U-shaped)

  • True or False?

    You can find the maximum or minimum value of A equals x squared plus 2 y by setting fraction numerator straight d A over denominator straight d x end fraction equals 0.

    False.

    You cannot find the maximum or minimum value of A equals x squared plus 2 y by setting fraction numerator straight d A over denominator straight d x end fraction equals 0 because A needs to be in terms of one variable only.

    You cannot have both x and y in the equation you want to maximise / minimise.

  • A rectangular field measures x m by y m with a perimeter of 100 m.

    Explain how you would find its area, A m2, in terms of x only.

    A rectangular field measures x m by y m with a perimeter of 100 m.

    To find its area, A m2, in terms of x only

    1. First find A in terms of x and y
      (A equals x y)

    2. Then use the perimeter information to find a relationship between x and y
      (2 x plus 2 y equals 100)

    3. Make y the subject (y equals 50 minus x) and substitute it into A

    This gives A equals x open parentheses 50 minus x close parentheses which expands to give A equals 50 x minus x squared. This is now in one variable only.

  • True or False?

    The minimum volume of the shape given by V equals x cubed minus 12 x plus 20 is the value of x that satisfies fraction numerator straight d V over denominator straight d x end fraction equals 0.

    False.

    The minimum volume of the shape given by V equals x cubed minus 12 x plus 20 is not the value of x that satisfies fraction numerator straight d V over denominator straight d x end fraction equals 0 as the minimum volume will be a value of V, not x.

    You need to find the two values of x that satisfies fraction numerator straight d V over denominator straight d x end fraction equals 0 then substitute them into the equation for V to get a maximum and a minimum value. The smaller value is the minimum value of V required.

  • What is a 'particle' in a kinematics question?

    In kinematics questions, the object that is moving is usually referred to as a 'particle'. It is modelled as taking up a single point in space.

  • Define displacement.

    Displacement is a vector quantity that measures how far a particle is from a fixed origin point.

    It can be positive or negative, and is usually measured in metres (m).

  • What is the difference between displacement and distance?

    Distance only measures 'how far', and is always a positive number.

    Displacement measures 'how far and in what direction', so in kinematics questions it can be positive or negative.

  • What letter is usually used for displacement in kinematics questions?

    Displacement is usually indicated by the letter s in kinematics questions.

    So the displacement of a particle at time t, for example, might be given as  s equals 3 t squared minus 2 t plus 1.

  • Define velocity.

    Velocity is a vector quantity that measures how fast a particle is traveling.

    It can be positive or negative, and is usually measured in metres per second (m/s).

  • What is the difference between velocity and speed?

    Speed only measures 'how fast', and is always a positive number.

    Velocity measures 'how fast and in what direction', so in kinematics questions it can be positive or negative.

  • If you know the equation for the displacement of a particle, how can you use calculus to find the velocity?

    Velocity is the derivative of displacement, so if you know the displacement equation for a particle you can differentiate it to find the the velocity.

    As an equation, that is  v equals fraction numerator straight d s over denominator straight d t end fraction.

  • What does a zero velocity value indicate?

    If velocity is zero, then the particle is stationary (not moving).

  • Define acceleration.

    Acceleration is a vector quantity that measures how fast the velocity is changing,

    It can be positive or negative, and is usually measured in metres per second squared (m/s2).

  • If you know the equation for the velocity of a particle, how can you use calculus to find the acceleration?

    Acceleration is the derivative of velocity, so if you know the velocity equation for a particle you can differentiate it to find the the acceleration.

    As an equation, that is  a equals fraction numerator straight d v over denominator straight d t end fraction.

  • What does a zero acceleration value indicate?

    If acceleration is zero, then the particle is moving at a constant velocity (i.e., its velocity is not changing).

  • What does "initial" or "initially" mean in the context of a kinematics question?

    "Initial" or "initially" refers to the conditions when t = 0 (at the start).