The gradient function of g \left( x \right) is g' \left( x \right) = 3. Sketch a possible graph of g \left( x \right).
Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.
Consider the graph of the gradient function f' \left( x \right):
What can be said about the graph of f \left(x\right)?
The gradient function, f' \left( x \right) is graphed:
Sketch a possible graph of f \left( x \right).
The gradient function, y' is graphed:
Sketch a possible graph of y.
The gradient function, g' \left( x \right) is graphed:
Sketch a possible graph of g \left( x \right).
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
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
- f \left( 0 \right) = - 3
- f \left( - 3 \right) = 0
- f \left( - 6 \right) = 1
- f' \left( - 6 \right) = 0
f' \left( x \right) < 0 for x > - 6
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( 2 \right) = 0
- f' \left( - 3 \right) = 0
f' \left( x \right) < 0 for - 3 < x < 2
f' \left( x \right) > 0 elsewhere
- 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
Sketch a quartic function, f \left( x \right), that satisfies the following conditions:
Sketch a function that matches the information for f \left( x \right):
- 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 - 2 < x < 2
f' \left( x \right) < 0 elsewhere
Consider the gradient function f' \left( x \right) = 2 x + 4 graphed below:
State the x-intercept of the gradient function.
What feature of f \left( x \right) does this x-intercept represent?
Is the gradient of f \left( x \right) for x > - 2 positive or negative?
Is the gradient of f \left( x \right) for x < - 2 positive or negative?
What kind of turning point is at the point \left( - 2 , - 1 \right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = - 2 x + 8 graphed below:
State the x-intercept of the gradient function.
Is the gradient of f \left( x \right) for x > 4 positive or negative?
Is the gradient of f \left( x \right) for x < 4 positive or negative?
What kind of turning point is at the point \left(4, 17\right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = 3 x - 6.
Graph the gradient function.
State the x-intercept of the gradient function.
What kind of stationary point is at the point \left(2, - 5 \right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = - 4 x + 8.
Graph the gradient function.
State the x-intercept of the gradient function.
What kind of stationary point is at the point \left(2, 11\right) on the graph of f \left( x \right)?
Consider the graph of the gradient function \dfrac{d y}{d x} = 4:
Find the equation of the function y if it has a y-intercept of 1.
Consider the gradient function \\ f' \left( x \right) = 2 \left(x + 3\right)^{2} graphed:
State the x-intercept of the gradient function.
Is the gradient of f \left( x \right) at values of x just to the left of - 3 positive or negative?
Is the gradient of f \left( x \right) at values of x just to the right of - 3 positive or negative?
What kind of feature is at the point \left( - 3 , 1\right) on the graph of f \left( x \right)?
Consider the gradient function \\ f' \left( x \right) = 3 \left(x - 6\right)^{2} graphed:
State the x-intercept(s) of the gradient function.
What kind of feature is at the point \left(6, 2\right) on the graph of f \left( x \right)?
Consider the gradient function \\ f' \left( x \right) = - 6 \left(x + 5\right)^{2} graphed:
State the x-intercept(s) of the gradient function.
What kind of feature is at the point \left( - 5 , 3\right) on the graph of f \left( x \right)?
Consider the gradient function \\ f' \left( x \right) = - 5 \left(x - 2\right)^{2} graphed:
State the x-intercept of the gradient function.
What kind of feature is at the point \left(2, 1\right) on the graph of f \left( x \right)?
The function f \left( x \right) has a derivative given by f' \left( x \right) = 6 \left(x + 7\right) \left(x + 2\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( - 7 , 52\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 2 , - 73 \right) on the graph of f \left( x \right)?
Consider the gradient function \\ f' \left( x \right) = - 6 \left(x - 1\right) \left(x - 5\right) graphed:
State the x-intercept(s) of the gradient function.
What kind of feature is at the point \left(1, - 13 \right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(5, 51\right) on the graph of f \left( x \right)?
Consider the gradient function \\ f' \left( x \right) = \left(x - 1\right)^{2} - 5 graphed:
State the coordinates of the turning point of f' \left( x \right).
Consider the gradient function f' \left( x \right) = - \left(x - 1\right)^{2} + 2.
Graph the gradient function.
State the coordinates of the turning point of f' \left( x \right).
What kind of feature is at the point \left(1, 4\right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = 6 \left(x + 6\right) \left(x + 2\right).
Graph the gradient function.
What kind of feature is at the point \left( - 6 , 2\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 2 , - 62 \right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = \left(x + 7\right)^{2} - 9.
Graph the gradient function.
State the coordinates of the turning point on the graph of the gradient function.
What kind of feature is at the point \left( - 7 , 64\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 10 , 82\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 4 , 46\right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = - \left(x - 3\right)^{2} + 9.
Graph the gradient function.
State the coordinates of the turning point on the graph of the gradient function.
What kind of feature is at the point \left(3, 30\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(0, 12\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(6, 48\right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = 12 \left(x + 2\right)^{2} \left(x + 5\right).
Graph the gradient function.
What kind of feature is at the point \left( - 5 , - 225 \right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 2 , - 144 \right) on the graph of f \left( x \right)?
Consider the gradient function f' \left( x \right) = 12 \left(x + 4\right) \left(x + 1\right) \left(x - 4\right):
What kind of feature is at the point \left( - 4 , - 256 \right) on the graph of f \left( x \right)?
What kind of feature is at the point \left( - 1 , 95\right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(4, - 1280 \right) on the graph of f \left( x \right)?
What kind of feature is at the point \left(2, - 688 \right) on the graph of f \left( x \right)?
Consider the following graph of y = f \left( x \right):
Sketch the graphs of y = f' \left( x \right) and \\ y = f'' \left( x \right).
Consider the following graph of y = f' \left( x \right):
Sketch the graph of y=f''(x).
For what intervals is the graph of \\ y = f \left( x \right) increasing?
For what interval is the graph of \\ y = f \left( x \right) concave up?
What are the x-values of the stationary points of y = f \left( x \right)?
Draw the gradient function for the following graphs:
The diagram shows the graph of y = f \left( x \right):
State the interval(s) where the values of the derivative f' \left( x \right) are negative.
What happens to f' \left( x \right) for large values of x?
Draw the graph of y = f' \left( x \right).