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Step 1: Sum the angles around the point and equate the sum to °. Step 2: Solve the equation to find . Now it’s your turn! If you get stuck, look back at the worked and guided examples for this section. Find angle . in each of the following diagrams: Worked Example. Find the value of angle x in the diagram below:
These angles are congruent angles located in the exterior region and on opposite sides of the transversal & at different vertices. List all pairs of alternate exterior angles in the diagram:
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5.3 Concavity and Inflection Points. Section 1: Determine the open intervals on which the graph s concave up or concave down. f (x) = 24 x2 + 12. f (x) = x2 + 1 x2 − 1. y = 2x − tan x π π , (− 2 ) Section 2: Find the inflection points and the open intervals of concavity. Section 3: Use the Second Derivative Test to find all relative ...
Angles Name: _____ Instructions • Use black ink or ball-point pen. • Answer all questions. • Answer the questions in the spaces provided – there may be more space than you need. • Diagrams are NOT accurately drawn, unless otherwise indicated. • You must show all your working out. Information
Example: y = x 3 − 6x 2 + 12x − 5. The derivative is: y' = 3x 2 − 12x + 12. The second derivative is: y'' = 6x − 12. And 6x − 12 is negative up to x = 2, positive from there onwards. So: f (x) is concave downward up to x = 2. f (x) is concave upward from x = 2 on. And the inflection point is at x = 2: Calculus Index.
Maxima and minima are also called TURNING points or STATIONARY points. Going from left to right, the gradient is DECREASING up to the point P and then it starts INCREASING again. The point P where the gradient stops decreasing and starts increasing is called an INFLECTION point.
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Example Determine where the function f(x)=x3 +3x2 +1 is increasing and decreasing, and where its graph is concave up and concave down. Find all relative extrema and points of inflection, and sketch the graph.
