뒤로Business Calculus: First Derivative Test, Increasing/Decreasing Intervals, and Local Extrema
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Q1. Given the graph below, answer the following:
A) Identify the intervals on which is increasing; Identify the interval on which .
B) Identify the intervals on which is decreasing; Identify the interval on which .
C) Find the -coordinate where has a local minimum.
D) Find the -coordinate where has a local maximum.
Background
Topic: First Derivative Test, Increasing/Decreasing Intervals, Local Extrema
This question tests your understanding of how to use the graph of a function to determine where it is increasing or decreasing, and to identify local minima and maxima (turning points) using the first derivative test.
Key Terms and Formulas:
Increasing Interval: is increasing on an interval if there.
Decreasing Interval: is decreasing on an interval if there.
Local Maximum: A point where changes from increasing to decreasing.
Local Minimum: A point where changes from decreasing to increasing.
Step-by-Step Guidance
Examine the graph and identify the segments where the curve is moving upwards as you go from left to right. These are the intervals where is increasing and .
Next, look for the segments where the curve is moving downwards as you go from left to right. These are the intervals where is decreasing and .
Find the points where the graph changes from decreasing to increasing. These are the -coordinates of local minima.
Find the points where the graph changes from increasing to decreasing. These are the -coordinates of local maxima.
List the intervals and points you identified, but do not write the final answers yet. Prepare to match the intervals to the labeled points (a, b, c, ...).
Try solving on your own before revealing the answer!
Final Answer:
A) Increasing intervals: From b to d, and from f to h ( on these intervals).
B) Decreasing intervals: From a to b, from d to f ( on these intervals).
C) Local minimum at: and .
D) Local maximum at: and .
These answers are based on where the graph changes direction, indicating local extrema and intervals of increase/decrease.
Q2. Given the graph below, answer the following:
A) Find the intervals on which is increasing.
B) Find the intervals on which is decreasing.
C) Find the relative minimum or local minimum.
D) Find the relative maximum or local maximum.
Background
Topic: First Derivative Test, Increasing/Decreasing Intervals, Local Extrema
This question is similar to Q1, focusing on interpreting a different graph to determine where the function is increasing, decreasing, and where it has local minima and maxima.
Key Terms and Formulas:
Increasing Interval: is increasing where .
Decreasing Interval: is decreasing where .
Local Minimum: Where the graph changes from decreasing to increasing.
Local Maximum: Where the graph changes from increasing to decreasing.
Step-by-Step Guidance
Look at the graph and identify the intervals where the curve rises as you move from left to right. These are the increasing intervals.
Identify the intervals where the curve falls as you move from left to right. These are the decreasing intervals.
Find the -coordinates where the graph switches from decreasing to increasing (local minima).
Find the -coordinates where the graph switches from increasing to decreasing (local maxima).
Prepare to match these intervals and points to the labeled positions (a, b, c, ...).
Try solving on your own before revealing the answer!
Final Answer:
A) Increasing intervals: From b to d, and from f to h.
B) Decreasing intervals: From a to b, from d to f.
C) Local minimum at: and .
D) Local maximum at: and .
These are determined by observing where the graph changes direction at the labeled points.
Q3. For the following functions, find the vertex (maximum or minimum point):
33.
34.
35.
36.
Background
Topic: Quadratic Functions, Vertex (Maximum/Minimum)
This question tests your ability to find the vertex of a quadratic function, which represents the maximum or minimum point of the parabola. For , the vertex occurs at .
Key Terms and Formulas:
Vertex of a parabola:
Maximum/Minimum Value: Plug the value of the vertex into to find the value.
If , the parabola opens upward (minimum); if , it opens downward (maximum).
Step-by-Step Guidance
For each function, identify the coefficients and .
Use the formula to find the -coordinate of the vertex.
Substitute this value back into the original function to find the corresponding value.
Determine whether the vertex is a maximum or minimum by checking the sign of .
Write down the vertex as for each function, but do not compute the final values yet.
Try solving on your own before revealing the answer!
Final Answer:
33. Vertex at (minimum, since )
34. Vertex at (minimum, since )
35. Vertex at (maximum, since )
36. Vertex at (maximum, since )
Each vertex is found using and substituting back into .
