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.
Consider the function y = e^{x} \left(x - 3\right).
Find the coordinates of the turning point.
State whether this is a minimum or maximum turning point.
Consider the function f \left( x \right) = 2 x^{3} - 12 x^{2} + 18 x + 3.
Find the x-coordinates of the turning points.
State the coordinates of the local maximum.
State the coordinates of the local minimum.
State the absolute maximum value on [0,7].
State the absolute minimum value on [0,7].
Consider the function f \left( x \right) = \dfrac{4 x^{9}}{\left(x + 2\right)^{4}}.
Find f' \left( 2 \right).
Is the function increasing or decreasing at x = 2?
Consider the function y = \dfrac{\sin x}{1 + \cos x}.
Find \dfrac{dy}{dx}.
State the number of turning points function y has.
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.
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.
Consider the curve f \left( x \right) = x^{3} - 5 x^{2} + 3 x - 5.
Find the x-coordinate(s) of the turning point(s).
Determine an equation for f'' \left( x \right).
Classify the stationary point(s).
Find the x-coordinates of the possible point of inflection, and verify that it is a point of inflection.
Sketch the graph of f \left( x \right) = x^{3} - 5 x^{2} + 3 x - 5.
Consider the curve y = x^{3} - 6 x^{2} - 3.
Find the coordinates of the stationary points.
Classify the stationary points.
Find the x-coordinates of the possible point of inflection, and verify that it is a point of inflection.
Sketch the graph of y = x^{3} + 6 x^{2} - 3.
Consider the function f \left( x \right) = \left(2 - 3 x\right)^{3} + 8.
State the coordinates of the y-intercept.
Find the the x-intercept(s).
Determine an equation for f' \left( x \right).
Hence find the x-coordinate(s) of the stationary point(s).
Classify the stationary point(s).
Sketch the graph of the function.
Consider the curve f \left( x \right) = \left(x^{2} - 9\right)^{2} + 3.
Find the coordinates of the turning points.
Classify the stationary points.
Find the x-coordinate(s) of any potential point(s) of inflection, and verify that they are point(s) of inflection.
Sketch the graph of f \left( x \right).
Consider the function f \left( x \right) = 4 e^{ - x^{2} }.
Find f' \left( x \right).
Find the values of x for which:
f' \left( x \right) \gt 0
f' \left( x \right) \lt 0
Find the limit of f(x) as x \to \infty.
Find the limit of f(x) as x \to - \infty.
Sketch the graph of f \left( x \right).
Consider the function f \left( t \right) = \dfrac{4}{2 + 3 e^{ - t }}.
Find f \left( 0 \right).
Find f' \left( t \right).
State whether f(x) is an increasing or decreasing function. Explain your answer.
Hence, state how many stationary points the function has.
Find the limit of f (t) as t \to \infty.
The graph of the function f \left( x \right) = e^{ 2 x} \sin 3 x is shown:
Find f' \left( x \right).
Find the x-intercepts, B and C.
Determine the coordinates of point A, correct to two decimal places.