11.11 Graphs and the derivative
Consider the graph of the function \\ f \left( x \right) = - \left(x - 4\right)^{3} + 7:
State the x-value of the stationary point of f \left( x \right).
State the domain where f \left( x \right) is decreasing.
State whether the rate of change of the following functions is positive, negative, or zero for all values of x:
For each of the following functions:
State the x-value of the stationary point of f \left( x \right).
State the domain where f \left( x \right) is increasing.
State the domain where f \left( x \right) is decreasing.
f \left( x \right) = - 2 \left(x + 2\right)^{2} + 4
f \left( x \right) = \left(x + 4\right) \left(x - 2\right)
f \left( x \right) = \dfrac{x^{3}}{3} - x-1
g \left( x \right) = - \left(x - 3\right)^{2} \left(x + 1\right)^{2}
f \left( x \right)
f \left( x \right)
Consider the function y = 4 x - 3.
Find the gradient function.
Sketch the graph of the gradient function.
For each of the following functions, sketch the gradient function:
y = x^{2}
f \left( x \right) = \left(x + 1\right)^{2}
y = x^{2} - 1
f \left( x \right) = x \left(x - 4\right) \left(x + 2\right)
Consider the function y = \left(x - 5\right)^{2} - 3 graphed below:
State the x-intercept of the gradient function.
For x \lt 5, state whether the values of the gradient function are above or below the x-axis.
For x \gt 5, state whether the values of the gradient function are above or below the x-axis.
Consider the function y = - \left(x + 7\right)^{2} + 5 graphed below:
State the x-intercept of the gradient function.
For x \lt - 7, state whether the values of the gradient function are above or below the x-axis.
For x > - 7, state whether the values of the gradient function are above or below the x-axis.
Consider the function y = - x^{2} - 10 x - 28 graphed below:
State the x-intercept of the gradient function.
For x \lt - 5, state whether the values of the gradient function are above or below the x-axis.
For x \gt - 5, state whether the values of the gradient function are above or below the x-axis.
Consider the function y = x^{2} + 10 x + 21 graphed below:
State the x-intercept of the gradient function.
For x \lt - 5, state whether the values of the gradient function are above or below the x-axis.
For x \gt - 5, state whether the values of the gradient function are above or below the x-axis.
Consider the function y = \left(x - 5\right)^{3} + 3:
State the coordinates of the point of inflection.
State the gradient of the curve at this point.
What feature does the gradient function have at x = 5?
For x \lt 5, state whether the values of the gradient function are above or below the x-axis.
For x \gt 5, state whether the values of the gradient function are above or below the x-axis.
What type of point is on the gradient function at x = 5?
Consider the function y = - \left(x + 7\right)^{3} - 3 :
State the coordinates of the point of inflection.
State the gradient of the curve at this point.
What feature does the gradient function have at x = - 7?
For x \lt - 7, state whether the values of the gradient function are above or below the x-axis.
For x \gt - 7, state whether the values of the gradient function are above or below the x-axis.
What type of point is on the gradient function at x = - 7?
Consider the graph of the gradient function f' \left( x \right):
What can be said about the graph of f \left(x\right)?
For each of the following gradient functions, sketch a graph of a possible original function:
State the type of point that matches the following descriptions:
A point where the curve changes from decreasing to increasing.
A point where the curve changes from increasing to decreasing.
A point where the tangent is horizontal and the concavity changes about the point.
For each of the following functions:
Find the derivative.
Find the coordinates of any stationary points.
Classify each stationary point.
f \left( x \right) = 134 - 300 x + 240 x^{2} - 64 x^{3}
f \left( x \right) = \left(x^{2} - 9\right)^{2} + 4
Consider the parabola with equation y = 5 + x - x^{2}.
Find the coordinates of the vertex of the parabola.
State the gradient of the tangent to the parabola at the vertex.
What type of stationary point is at the vertex of this parabola?
Consider the function f \left( x \right) = x^{2} + 4 x + 9.
Find an equation for the gradient function f' \left( x \right).
State the interval in which the function is increasing.
State the interval in which the function is decreasing.
Find the coordinates of the stationary point.
Classify the stationary point.
Consider the function f \left( x \right) = 4 x^{3} + 5 x^{2} - 4.
Find an equation for the gradient function f' \left( x \right).
Find the coordinates of the stationary points.
Complete the table of values:
| x | -2 | -\dfrac{5}{6} | -\dfrac{1}{2} | 0 | 1 |
|---|---|---|---|---|---|
| f'\left( x \right) | 0 | 0 |
Hence determine the:
Local minimum
Local maximum
Is - 4 the absolute minimum value of the function? Explain your answer.
Consider the function f \left( x \right) = 3 x^{2} - 54 x + 241.
Find f' \left( x \right).
Find the x-coordinate of the stationary point.
Classify the stationary point.
Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.
Sketch the quadratic function, f \left( x \right), that satisfies the following conditions:
- f \left( 0 \right) = - 18
- f \left( 3 \right) = 0
- f \left( 6 \right) = 6
- f' \left( 6 \right) = 0
f' \left( x \right) > 0 for x < 6
- f \left( 0 \right) = 16
- f \left( 2 \right) = 0
- f \left( 5 \right) = - 9
- f' \left( 5 \right) = 0
f' \left( x \right) < 0 for x < 5
- f \left( 0 \right) = 5
- f \left( - 2 \right) = 0
- f' \left( 3 \right) = 0
f' \left( x \right) > 0 for x < 3
f \left( 0 \right) = 10
f \left( - 2 \right) = 0
f \left( - 6 \right) = - 8
f' \left( - 6 \right) = 0
f' \left( x \right) < 0 for x < - 6
Sketch the cubic function, f \left( x \right), that satisfies the following conditions:
- f \left( 0 \right) = 7
- f \left( - 2 \right) = 0
- f \left( - 4 \right) = - 1
- f' \left( - 4 \right) = 0
f' \left( x \right) > 0 for x < - 4
f' \left( x \right) > 0 for x > - 4
- f' \left( 2 \right) = 0
- f' \left( - 3 \right) = 0
f' \left( x \right) < 0 for - 3 < x < 2
f' \left( x \right) > 0 elsewhere
Sketch the quartic function, f \left( x \right), that satisfies the following conditions:
- f' \left( - 1 \right) = 0
- f' \left( 4 \right) = 0
f' \left( x \right) > 0 for x > 4
f' \left( x \right) < 0 elsewhere
- f \left( 0 \right) = 0
- f' \left( 0 \right) = 0
- f' \left( 2 \right) = 0
- f' \left( - 2 \right) = 0
f' \left( x \right) > 0 for x < - 2, 0 < x < 2
f' \left( x \right) < 0 elsewhere
Consider the equation of the parabola y = 3 x^{2} - 18 x + 24.
Find the x-intercepts.
Find the y-intercept.
Find \dfrac{dy}{dx}.
Find the stationary point.
Classify the stationary point.
Sketch the graph of the parabola.
For each of the following functions:
Find the y-intercept.
Find the x-intercepts.
Find f' \left( x \right).
Hence find the x-coordinates of the stationary points.
Classify the stationary points.
Sketch the graph of the function.
Sketch the graph of the following functions showing all stationary points: