9. Practice on curve sketching › The suggested procedures of sketching the graph of a function are as follows: 1. List out the general properties of the function; i.e., intercepts, symmetry, asymptotes, domain, and range. 2. Perform the 1 st derivative test to find critical points. Determine the trend of the graph in different intervals. 3. Perform the 2 nd derivative test to find inflection points and determine the concavity in different intervals. Use the results to confirm relative extrema found in step 2. CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 23
Transcript
9. Practice on curve sketching
› The suggested procedures of sketching the graph of a function are as follows:
1. List out the general properties of the function; i.e., intercepts, symmetry, asymptotes, domain, and range.
2. Perform the 1st derivative test to find critical points. Determine the trend of the graph in different intervals.
3. Perform the 2nd derivative test to find inflection points and determine the concavity in different intervals. Use the results to confirm relative extrema found in step 2.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 23
› Example: Sketch .› Solution: We first look at the general properties of .› Domain: › Range: › The -intercept: › The -intercepts: We need to solve the polynomial equation:
› Hence, , , [ is a triple root]
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 24
› The end behavior: From QIII to QI› Asymptotes: None› Now we can perform the 1st derivative test:
› The critical values are therefore and .
› Then check the trend of in different intervals:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 25
Increasing Decreasing Decreasing Increasing
› We know that then is a maximum and is a minimum. But we don’t know what is .
› To answer this question, we perform the 2nd derivative test.
› Since , is a stationary, inflection point.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 26
› The other two points will be typical inflection points where the function changes its concavity.
› Using this information, we would be able to sketch the graph of .
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 27
Concave down Concave up Concave down Concave up
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 28
Maximum
MinimumInflection points
› Example: Sketch
› Solution: We first find out the domain of the function. Note that gives . Hence
› This also tells us that there are two vertical asymptotes:
› This is a rational function with .
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 29
› Therefore,
› Now we can check for intercepts. Setting yields
› Setting gives
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 30
→ →
› To find the critical values, we differentiate :
› When , . Hence there is only one stationary point. Meanwhile, there are two vertical asymptotes which are critical points too.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 31
Decreasing Decreasing Increasing Increasing
› Take the second derivative of :
› There are two inflection points, , which correspond to the vertical asymptotes.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 32
› To check the concavity:
› Based on these, we can conclude that should be a relative minimum. There exists no relative maximum.
› Moreover, we can deduce that the range of should be as follows:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 33
Concave down Concave up Concave down
› Finally, we can plot the graph.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 34
› Example: Sketch the graph of .
› Solution: Note that the function is undefined when . That means
› There exists one vertical asymptote:
› This function does not have -intercept since it is undefined when .
› Similarly, it has no -intercept since has no solutions.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 35
› When ,
› Hence, it has an oblique asymptote whose equation is given by
› The graph is based on with an oblique asymptote of .
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 36
→± →±
Oblique asymptote:
› The first derivative of :
› The critical values are .› When , . Hence, the critical points are .› Now we can check the trends:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 37
Increasing Decreasing Decreasing Increasing
› The second derivative of :
› Since for all , yet makes undefined, this function has no inflection point.
› Now check for concavity:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 38
Concave down Concave up
› Using the information we can plot the graph.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 39
› Example: Sketch / / .› Solution: This function is defined everywhere; therefore its domain is . It implies there is no vertical asymptote.
› The -intercept is:
› To find the -intercepts it needs a little bit trick:
› When , . CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION
(PART 2) 40
/
/ /
/
› Since / increases faster than / , when , .
› Now find the first derivative of :
› Setting yields
› It gives a stationary point of 2. Note that makes undefined. It is therefore a critical point.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 41
/ /
/
› We can check the trends:
› Then we look for the concavity. First we deduce the second derivative:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 42
Increasing Decreasing Increasing
/ /
/
› Setting :
› We can determine the inflection points: .› Using these we can work out the concavity:
› Compared with the trend table, we know that is a local minimum, yet is a cusp.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 43
/
Concave down Concave up Concave up
› Finally we can sketch the graph:
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 44
› Practice: Sketch the graph for the following functions.
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 45
/ /
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 46
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 47
/ /
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 48
CHAPTER 4 - APPLICATIONS OF DIFFERENTIATION (PART 2) 49
