We use the first derivative test to identify where a function has a local max and where it has a local minimum. We must identify where the function is increasing (where f'(x) > 0) and where the function is decreasing (where f'(x) < 0). The worksheet below gives practice for those skills.
In a Business Calculus class on Oct. 22, 2020, the day we hit max and mins, I had the following thoughts.[br][br]There are lots of songs about Hills and Valleys. I did a search and found some to link to. I put them here for your encouragement for the times when you need such encouragement, or as a resource in case you need ideas to encourage others. [url=https://www.biblegateway.com/quicksearch/?quicksearch=encourage&version=NASB]1 Thessalonians 5:11[/url], Therefore encourage one another and build up one another, just as you are also doing.[br][br]The fact that there are so many songs like this shows how many people need encouragement. If you look at many of the comments with these videos, you'll see they have brought encouragement to many. [br]Hills and Valleys by Tauren Wells-[url=https://www.youtube.com/watch?v=iGyEO80Ga1o]https://www.youtube.com/watch?v=iGyEO80Ga1o[/url][br]God on the Mountain by Lynda Randle-[url=https://www.youtube.com/watch?v=HUBY9TxnAMw]https://www.youtube.com/watch?v=HUBY9TxnAMw[/url] (others have done this song as well).[br]Highlands (Song of Ascent) by Hillsong United: [url=https://www.youtube.com/watch?v=2UcqNYaduo0]https://www.youtube.com/watch?v=2UcqNYaduo0[/url][br]I will praise you in this Valley by The Browders [url=https://www.youtube.com/watch?v=T67slRZ4hSo]https://www.youtube.com/watch?v=T67slRZ4hSo[/url][br]I will praise you by Elga Samuel [url=https://www.youtube.com/watch?v=PKB2hc6F7T8]https://www.youtube.com/watch?v=PKB2hc6F7T8[/url][br]It is Well with My Soul (this version by Vineyard Worship) [url=https://www.youtube.com/watch?v=X6yDFn3OAFo]https://www.youtube.com/watch?v=X6yDFn3OAFo[/url][br]Symphony by Switch [url=https://www.youtube.com/watch?v=C_90_NAbv3k]https://www.youtube.com/watch?v=C_90_NAbv3k[/url] (this is more a valley song, even though that word isn't used)[br]Up Again by Dan Bremnes [url=https://www.youtube.com/watch?v=GmewkjWe3Bw]https://www.youtube.com/watch?v=GmewkjWe3Bw[/url] (again, doesn't use the words hills and valleys, but does about successes and failures)[br][br]Psalm 23,[url=https://www.biblegateway.com/passage/?search=psalms+23&version=NASB]https://www.biblegateway.com/passage/?search=psalms+23&version=NASB[/url] , also talks about the highs and lows of life.[br][br]I offered extra credit if students shared another favorite hill/valley related scripture, song or poem. What you see below are items shared by my students.[br][br][url=https://biblia.com/bible/nasb95/1%20Sam%202.7-8]1 Samuel 2:7-8[/url] “The Lord makes poor and rich; He brings low, He also exalts. He raises the poor from the dust, He lifts the needy from the ash heap to make them sit with nobles, and inherit a seat of honor; For the pillars of the earth are the Lord’s, and He set the world on them.”[br][br]One of my favorite quotes ever is by, Richard M. Nixon, “Only if you have been in the deepest valley, can you ever know how magnificent it is to be on the highest mountain,” I love this because it shows, without struggling and feeling at your lowest, you will never experience success or being great. If everything was given to you, and you never struggled, you would never experience the greatest high(mountain) in the same way, or respect it as much. It shows struggle is good.[br][br]This song is one of my favorites, and I feel like it does not literally refer to hills and valleys, but figuratively. With the positives (hills) of being a disciple to Jesus, and then failing him (valleys).[br][url=https://www.youtube.com/watch?v=BRkdIx1YDso&ab_channel=LegacyFive]https://www.youtube.com/watch?v=BRkdIx1YDso&ab_channel=LegacyFive[br][/url][br]I really enjoyed in business calculus how you incorporated the spiritual hills and valleys in our discussion on graphing derivatives. Last Sunday during my church’s worship service we sang Graves into Gardens by Elevation Worship ([url=https://www.youtube.com/watch?v=kTvv-9zosPs]https://www.youtube.com/watch?v=kTvv-9zosPs[/url]). One of the lyrics of the song says, “Cause the God of the mountain is the God of the valley. There’s not a place Your mercy and grace won’t find me again.” This lyric reminded me of our class discussion and how God is always in control throughout all of the highs and lows of our lives.[br][br]I wanted to share this song called "Who Am I". My family and I have been listening to this song for years and it has been a very encouraging song in our lives. Who Am I by Casting Crowns- [url=https://www.youtube.com/watch?reload=9&v=3rT8Re1EIQc]https://www.youtube.com/watch?reload=9&v=3rT8Re1EIQc[br][/url]
In a Business Calculus class on Oct. 22, 2020, the day we hit max and mins, I had the following thoughts.[br][br]There are lots of songs about Hills and Valleys. I did a search and found some to link to. I put them here for your encouragement for the times when you need such encouragement, or as a resource in case you need ideas to encourage others. [url=https://www.biblegateway.com/quicksearch/?quicksearch=encourage&version=NASB]1 Thessalonians 5:11[/url], Therefore encourage one another and build up one another, just as you are also doing.[br][br]The fact that there are so many songs like this shows how many people need encouragement. If you look at many of the comments with these videos, you'll see they have brought encouragement to many. [br]Hills and Valleys by Tauren Wells-[url=https://www.youtube.com/watch?v=iGyEO80Ga1o]https://www.youtube.com/watch?v=iGyEO80Ga1o[/url][br]God on the Mountain by Lynda Randle-[url=https://www.youtube.com/watch?v=HUBY9TxnAMw]https://www.youtube.com/watch?v=HUBY9TxnAMw[/url] (others have done this song as well).[br]Highlands (Song of Ascent) by Hillsong United: [url=https://www.youtube.com/watch?v=2UcqNYaduo0]https://www.youtube.com/watch?v=2UcqNYaduo0[/url][br]I will praise you in this Valley by The Browders [url=https://www.youtube.com/watch?v=T67slRZ4hSo]https://www.youtube.com/watch?v=T67slRZ4hSo[/url][br]I will praise you by Elga Samuel [url=https://www.youtube.com/watch?v=PKB2hc6F7T8]https://www.youtube.com/watch?v=PKB2hc6F7T8[/url][br]It is Well with My Soul (this version by Vineyard Worship) [url=https://www.youtube.com/watch?v=X6yDFn3OAFo]https://www.youtube.com/watch?v=X6yDFn3OAFo[/url][br]Symphony by Switch [url=https://www.youtube.com/watch?v=C_90_NAbv3k]https://www.youtube.com/watch?v=C_90_NAbv3k[/url] (this is more a valley song, even though that word isn't used)[br]Up Again by Dan Bremnes [url=https://www.youtube.com/watch?v=GmewkjWe3Bw]https://www.youtube.com/watch?v=GmewkjWe3Bw[/url] (again, doesn't use the words hills and valleys, but does about successes and failures)[br][br]Psalm 23,[url=https://www.biblegateway.com/passage/?search=psalms+23&version=NASB]https://www.biblegateway.com/passage/?search=psalms+23&version=NASB[/url] , also talks about the highs and lows of life.[br][br]I offered extra credit if students shared another favorite hill/valley related scripture, song or poem. What you see below are items shared by my students.[br][br][url=https://biblia.com/bible/nasb95/1%20Sam%202.7-8]1 Samuel 2:7-8[/url] “The Lord makes poor and rich; He brings low, He also exalts. He raises the poor from the dust, He lifts the needy from the ash heap to make them sit with nobles, and inherit a seat of honor; For the pillars of the earth are the Lord’s, and He set the world on them.”[br][br]One of my favorite quotes ever is by, Richard M. Nixon, “Only if you have been in the deepest valley, can you ever know how magnificent it is to be on the highest mountain,” I love this because it shows, without struggling and feeling at your lowest, you will never experience success or being great. If everything was given to you, and you never struggled, you would never experience the greatest high(mountain) in the same way, or respect it as much. It shows struggle is good.[br][br]This song is one of my favorites, and I feel like it does not literally refer to hills and valleys, but figuratively. With the positives (hills) of being a disciple to Jesus, and then failing him (valleys).[br][url=https://www.youtube.com/watch?v=BRkdIx1YDso&ab_channel=LegacyFive]https://www.youtube.com/watch?v=BRkdIx1YDso&ab_channel=LegacyFive[br][/url][br]I really enjoyed in business calculus how you incorporated the spiritual hills and valleys in our discussion on graphing derivatives. Last Sunday during my church’s worship service we sang Graves into Gardens by Elevation Worship ([url=https://www.youtube.com/watch?v=kTvv-9zosPs]https://www.youtube.com/watch?v=kTvv-9zosPs[/url]). One of the lyrics of the song says, “Cause the God of the mountain is the God of the valley. There’s not a place Your mercy and grace won’t find me again.” This lyric reminded me of our class discussion and how God is always in control throughout all of the highs and lows of our lives.[br][br]I wanted to share a verse with you in regards to hills and valleys. 1 Kings 20:28 says:[br]"And a man of God came near and said to the king of Israel, “Thus says the Lord, ‘Because the Syrians have said, “The Lord is a god of the hills but he is not a god of the valleys,” therefore I will give all this great multitude into your hand, and you shall know that I am the Lord.’”[br][br]I wanted to share this song called "Who Am I". My family and I have been listening to this song for years and it has been a very encouraging song in our lives. Who Am I by Casting Crowns- [url=https://www.youtube.com/watch?reload=9&v=3rT8Re1EIQc]https://www.youtube.com/watch?reload=9&v=3rT8Re1EIQc[br][/url]
Psalms 104:8 The mountains rose, the valleys sank down to the place that you appointed for them.[br][br]Do it again, by Elevation Worship, https://www.youtube.com/watch?v=ZOBIPb-6PTc.[br][br]Psalm 23:4 “Even though I walk through the valley of shadow of death, I fear no evil, for You are with me; Your rod and Your staff, they comfort me.”