text: iGCSE (2021 Edition)
  
  # 11.01 Tangents and the derivative
  
  Worksheet
  Tangents and the derivative
  1
  
  For each of the following, find the gradient of the tangent at the given point:
  
  a
  b
  c
  d
  e
  f
  g
  2
  
  Consider the graph of y = x below:
  
  a
  
  Find the gradient of the line at x = 4.
  
  b
  
  Find the gradient at any value of x.
  
  c
  
  True or false: A linear function has a constant gradient.
  
  3
  
  Find the gradient of the tangent to the following functions for any value of x:
  
  a
  
  f(x) = 1
  
  b
  
  f(x) = 2x - 4
  
  4
  
  For each of the following exponential functions, use the graph to find the gradient of the tangent drawn at x = 0:
  
  a
  b
  5
  
  Consider the following graph of the function f \left( x \right) = - \left(x - 4\right)^{3} + 7:
  
  a
  
  State the x-value of the stationary point of f \left( x \right).
  
  b
  
  State the region(s) of the domain where f \left( x \right) is decreasing.
  
  6
  
  For each graph of the following functions:
  
  i
  
  Find the x-value of any stationary point(s) of f \left( x \right).
  
  ii
  
  State the region(s) of the domain where f \left( x \right) is increasing.
  
  iii
  
  State the region(s) of the domain where f \left( x \right) is decreasing.
  
  a
  b
  c
  d
  e
  f
  g
  7
  
  Consider the graph of function f \left( x \right):
  
  a
  
  Is there a stationary point of f \left( x \right) when x = 0?
  
  b
  
  Find the x-value of the stationary point of f \left( x \right) when x < 0.
  
  c
  
  Over what region in the domain is f \left( x \right) constant?
  
  d
  
  State the region of the domain where f \left( x \right) is increasing.
  
  e
  
  State the region of the domain where f \left( x \right) is decreasing.
  
  8
  
  Consider the following figure. The black secant line through the given point has the equation y = 2.02 x - 9.14, while the purple secant line through the point has the equation y = 2.01 x - 9.07.
  
  a
  Find the coordinates of point A
  b
  
  Estimate the gradient of the tangent at point A, to the nearest integer.
  
  9
  
  Consider the graph of the function f \left( x \right) and the tangent at x=0 graphed below:
  
  Complete the table of values:
  
  10
  
  Consider the graph of the cubic function f \left( x \right) and the tangents at x=-3, x=0 and x=3 graphed below:
  
  a
  
  Complete the following table of values:
  
  b
  
  Use the set of data points to draw a possible graph of y = f' \left( x \right).
  
  11
  
  Determine whether rate of change of the following functions is positive, negative or zero for all values of x:
  
  a
  
  f(x) = -8
  
  b
  
  f(x) = 4x-1
  
  c
  
  f(x) = -x+10
  
  12
  
  Consider the function f \left( x \right) = x^2 along with its gradient function graphed below:
  
  a
  
  Which feature of the gradient function tells us whether f \left( x \right) = x^{2} is increasing or decreasing?
  
  b
  
  For x > 0, is the gradient of the tangent positive or negative?
  
  c
  
  For x \geq 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?
  
  d
  
  For x < 0, is the gradient of the tangent positive or negative?
  
  e
  
  For x < 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?
  
  f
  
  For f \left( x \right) = x^{2}, the derivative f' \left( x \right) is what type of function?
  
  13
  
  Consider the function f \left( x \right) = x^3 along with its gradient function graphed below:
  
  a
  
  For x > 0, is the gradient of the tangent positive or negative?
  
  b
  
  For x \geq 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?
  
  c
  
  For x < 0, is the gradient of the tangent positive or negative?
  
  d
  
  For x < 0, as the value of x increases, does the gradient of the tangent line decrease at an decreasing rate, decrease at a constant rate, remain constant?
  
  e
  
  For f \left( x \right) = x^{3}, the derivative f' \left( x \right) is what type of function?
  
  14
  
  Sean draws the graphs of x^{2}, x^{3}, x^{4}, and x^{5} and draws the tangents to each one at the point where x = 1. Sean then notes where each of the tangents cut the y-axis and records this in the table below:
  
  a
  Complete the table using the given information to calculate the gradient of each of the tangents at x = 1.
  b
  
  Following the pattern in the table, what would be the gradient of the tangent to the graph of f \left( x \right) = x^{n} at the point where x = 1?
  
  c
  
  Could the equation of the derivative of f \left( x \right) = x^{6} be f' \left( x \right) = x^{5}? Explain your answer.