11.08 Using the first derivative
Determine whether following statements describe a maximum or a minimum turning point:
A point where the curve changes from increasing to decreasing.
A point where the curve changes from decreasing to increasing.
Consider the function
f \left( x \right) = - \left(x - 4\right)^{3} + 7 shown:
What is the x-coordinate of the stationary point?
State the domain for which f \left( x \right) is decreasing.
For each of the functions graphed below:
State the x-coordinate of the stationary point(s).
State the domain for which f \left( x \right) is increasing.
State the domain for which f \left( x \right) is decreasing.
Consider the function f \left( x \right) shown:
State the x-coordinate of the stationary point.
State the domain for which f \left( x \right) is increasing.
State the domain for which f \left( x \right) is decreasing.
State the domain for which f \left( x \right) is constant.
For each of the following functions:
Find the derivative.
Find the coordinates of any stationary points.
Classify the stationary points.
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 coordinates of any stationary points.
Classify the stationary point.
Sketch the graph of the parabola.
For each of the following quadratic equations:
Find the coordinates of the turning point.
Classify the turning point.
Consider the function f \left( x \right) = 134 - 300 x + 240 x^{2} - 64 x^{3}.
Find the derivative.
Find the coordinates of the stationary point.
Classify the stationary point.
Consider the function f \left( x \right) = \left(x^{2} - 9\right)^{2} + 4.
Find the derivative.
Find the coordinates of the stationary points.
Classify the stationary points.
Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.
Sketch a 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
Sketch a cubic function, f \left( x \right), that satisfies the following conditions:
- f' \left( - 5 \right) = 0
f' \left( x \right) > 0 for all other values of x.
- 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 a 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
The function y = a x^{2} - b x + c passes through the points (5, - 42) and (4, - 66) and has a maximum turning point at x = 3.
Find \dfrac{dy}{dx}.
Find the value of a
Find the value of c.
Find the value of b.
Consider the cubic function y = x^{3} - a x^{2} + b x + 11, which has stationary points at x=2 and x=10.
Find \dfrac{dy}{dx}.
Find the value of a.
Hence find the value of b.
Consider the gradient function f' \left( x \right) = 2 x + 6 graphed below:
State the x-intercept of the gradient function.
Is the gradient of f \left( x \right) for x > - 3 positive or negative?
Is the gradient of f \left( x \right) for x < - 3 positive or negative?
What feature of f \left( x \right) does this x-intercept represent?
The function f \left( x \right) has a derivative given by f' \left( x \right) = 6 \left(x - 2\right) \left(x - 7\right). A graph of the derivative function is shown:
State the x-intercept(s) of the gradient function.
What kind of feature is at the point \left(2, 77\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(7, - 48 \right) on the graph of f \left( x \right)?
The function f \left( x \right) has a derivative given by f' \left( x \right) = 3 \left(x + 5\right)^{2}. A graph of the derivative function is shown:
State the x-intercept(s) of the gradient function.
What kind of feature is at the point \left( - 5 , 2\right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = 12 \left(x + 4\right)^{2} \left(x + 7\right).
Graph the gradient function.
What kind of feature is at the point \left( - 7 , - 1617 \right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 4 , - 1536 \right) on the graph of f \left( x \right)?