Unit 4 - Differential Calculus
Thinking Points
Given a point \(x=c\), what can we say about the graph of the function in the following cases.
+ \(f(c) = 0\)
The graph of \(y=f(x)\) intersects the x-axis at \(x = c\). The point \((c, 0)\) is an x-intercept (or root) of the function.
+ \(f(c) > 0\)
The graph of \(y=f(x)\) is located strictly above the x-axis at \(x = c\).
+ \(f'(c) = 0\)
The graph of \(y=f(x)\) has a horizontal tangent at \(x = c\). This is a stationary point, which could be a local maximum, a local minimum, or a stationary point of inflection.
+ \(f'(c) < 0\)
The graph of \(y=f(x)\) is strictly decreasing at \(x = c\). The tangent line to the curve at this point has a negative slope.
+ \(f''(c) > 0\)
The graph is concave up (bending upwards) at \(x = c\). The tangent at \(x=c\) lies below the curve.
+ \(f'(c) = 0\) and \(f''(c) < 0\)
Since \(f'(c)=0\), there is a stationary point at \(x=c\). By the Second Derivative Test, the graph has a local maximum at \(x = c\). The tangent is horizontal and the curve is concave down.
+ \(f(c) = 0\) and \(f'(c) > 0\) and \(f''(c) = 0\)
At \(x = c\), the graph crosses the x-axis (x-intercept) while strictly increasing. Furthermore, because the second derivative is zero, \((c, 0)\) is a possible point of inflection where the concavity of the graph may be changing.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Describe in your own words what the value of \(f\left(x\right)\) tells us about the graph.
Describe in your own words what the value of \(f’\left(x\right)\) tells us about the graph.
Describe in your own words what the value of \(f’’\left(x\right)\) tells us about the graph.
Common Mistakes / Misconceptions
It is common to forget that \(f\left(x\right)\gt 0\) means the function is positive and above the \(x\)-axis at that point, or to confuse this with the idea of increasing.
Connecting This to Other Skills
This concept builds on the foundations we have built throughout Differential Calculus (Unit 4)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?