Transcript Exponential Functions PowerPoint

EXPONENTIAL GROWTH
•WHEN HAVE YOU SEEN EXPONENTIAL GROWTH IN A REAL-WORLD
SITUATION?
•WHAT ARE SOME PROPERTIES OF EXPONENTIAL GROWTH
FUNCTIONS?
•DOMAIN/RANGE, INTERCEPTS, CONTINUITY, END BEHAVIOR, ETC.
EXPONENTIAL DECAY
•WHEN HAVE YOU SEEN EXPONENTIAL DECAY IN A REAL-WORLD
SITUATION?
•WHAT ARE SOME PROPERTIES OF EXPONENTIAL DECAY
FUNCTIONS?
• DOMAIN/RANGE, INTERCEPTS, CONTINUITY, END BEHAVIOR, ETC.
EXPONENTIAL GROWTH AND DECAY
•EXPONENTIAL GROWTH AND DECAY FORMULA:
•𝑁 =
𝑁0 (1 + 𝑟)𝑡
• WHERE N˳ IS THE INITIAL AMOUNT
• R IS THE GROWTH RATE IF R > 0
• R IS THE DECAY RATE IF R < 0
• T IS THE AMOUNT OF TIME
EXPONENTIAL GROWTH AND DECAY
•CONTINUOUS EXPONENTIAL GROWTH AND DECAY FORMULA:
•𝑁 =
𝑁0 𝑒 𝑘𝑡
• WHERE N˳ IS THE INITIAL AMOUNT
• K IS THE CONTINUOUS GROWTH RATE, THEN K > 0
• K IS THE CONTINUOUS DECAY RATE, THEN K < 0
• T IS THE AMOUNT OF TIME
GROWTH OR DECAY
•MEXICO HAS A POPULATION OF APPROXIMATELY 110 MILLION PEOPLE.
IF MEXICO’ S POPULATION CONTINUES TO GROW AT THE DESCRIBED
RATE, PREDICT THE POPULATION OF MEXICO IN 10 AND 20 YEARS.
• IF GROWTH RATE IS 1.42% ANNUALLY
• IF GROWTH RATE IS 1.42% CONTINUOUSLY
GROWTH OR DECAY
•THE POPULATION OF A TOWN IS DECLINING AT A RATE OF 6%. IF
THE CURRENT POPULATION IS 12,426 PEOPLE, PREDICT THE
POPULATION IN 5 AND 10 YEARS USING:
•ANNUALLY
•CONTINUOUSLY
EXAMPLES
•JASMINE RECEIVES A 3.5% RAISE AT THE END OF EACH YEAR FROM
HER EMPLOYER TO ACCOUNT FOR INFLATION. WHEN SHE STARTED
WORKING FOR THE COMPANY IN 1994, SHE WAS EARNING A SALARY
OF $31,000.
• WHAT WAS JASMINES SALARY IN 2000? 2009?
• WHAT WILL IT BE IN 2024?
EXAMPLES
•THE HALF-LIFE OF A RADIOACTIVE SUBSTANCE IS THE AMOUNT OF
TIME IT TAKES FOR HALF OF THE ATOMS OF THE SUBSTANCE TO
DISINTEGRATE. URANIUM-235 IS USED TO FUEL A COMMERCIAL
POWER PLANT. ITS HALF-LIFE IS 704 MILLION YEARS.
•HOW MANY GRAMS OF URANIUM WILL REMAIN AFTER 1 MILLION
YEARS IF YOU START WITH 200 GRAMS?
EXAMPLES
•UNDER THE RIGHT GROWING CONDITIONS A PARTICULAR
SPECIES OF PLANT HAS A DOUBLING TIME OF 12 DAYS. SUPPOSE A
PASTURE CONTAINS 46 PLANTS OF THIS SPECIES. HOW MANY
PLANTS WILL THEIR BE AFTER 20, 65, AND X DAYS?
TRUE OR FALSE
•EXPONENTIAL FUNCTIONS CAN NEVER HAVE RESTRICTIONS ON THE
DOMAIN.
•EXPONENTIAL FUNCTIONS ALWAYS HAVE RESTRICTIONS ON THE
RANGE
•GRAPHS OF EXPONENTIAL FUNCTIONS ALWAYS HAVE AN ASYMPTOTE.
EXTRA PRACTICE
•WORKSHEET
M&M ACTIVITY
•FOLLOW THE DIRECTIONS ON THE WORKSHEET.
COMPOUND INTEREST
COMPOUND INTEREST
•FORMULA:
• A 𝑡 = 𝑃(1 +
•WHERE P = INITIAL AMOUNT INVESTED, R = INTEREST RATE,
𝑟 𝑛𝑡
)
𝑛
T = TIME
COMPOUND INTEREST
•CONTINUOUS COMPOUND FORMULA:
•𝐴 = 𝑃𝑒 𝑟𝑡
• USED ONLY WHEN DOING COMPOUNDED DAILY!
COMPOUND INTEREST
•WHAT ARE THE DIFFERENT TYPES OF COMPOUNDING?
• ANNUALLY
• SEMI-ANNUALLY
• QUARTERLY
• MONTHLY
• WEEKLY
• DAILY
COMPOUND INTEREST
•IF I INVESTED $500 INTO AN ACCOUNT WHICH YIELDS 5% ANNUAL
INTEREST, HOW MUCH WOULD I HAVE AFTER 7 YEARS IF THE INTEREST
WAS COMPOUNDED
• ANNUALLY
• QUARTERLY
• DAILY
COMPOUND INTEREST
• KRYSTI INVESTS $1200 IN AN ACCOUNT WITH A 6% INTEREST RATE, MAKING
NO OTHER DEPOSITS OR WITHDRAWLS. WHAT WILL KRYSIT’S ACCOUNT
BALANCE BE AFTER 20 YEARS IF THE INTEREST IS COMPOUNDED:
• SEMI-ANNUALLY
• MONTHLY
• DAILY
COMPOUND INTEREST
•WHICH SHOULD I INVEST $1200 IN?
•A 3 YEAR CD WITH 3.45% INTEREST COMPOUNDED
CONTINUOUSLY
•A 5 YEAR CD WITH 4.75% INTEREST COMPOUNDED MONTHLY?