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5.3 Increasing and Decreasing Intervals
Calculus
The following graphs show the derivative of 𝒇 , 𝒇ᇱ. Identify the intervals when 𝒇 is increasing and decreasing. Include a justification statement.
Increasing:
Decreasing:
Increasing:
Decreasing:
For each function, find the intervals where it is increasing and decreasing, and JUSTIFY your conclusion. Construct a sign chart to help you organize the information, but do not use a calculator.
- 𝑓ሺ𝑥ሻ ൌ 𝑥 ଷ^ െ 12 𝑥 1 4. 𝑔ሺ𝑥ሻ ൌ 𝑥 ଶ^ ሺ𝑥 െ 3 ሻ
- 𝑓ሺ𝑥ሻ ൌ 𝑥 ଶ^ 𝑒 ௫^ 6. 𝑔ሺ𝑡ሻ ൌ 12 ሺ 1 cos 𝑡ሻ on the interval ሺ0, 2𝜋ሻ
Practice
x
y
𝒇ᇱ^ ሺ𝒙ሻ
The derivative 𝒇ᇱ^ is given for each problem. Use a calculator to help you answer each question about 𝒇.
- 𝑓ᇱ^ ሺ𝑥ሻ ൌ ௫ାଷ^
షೣ ௫ మ^ ା.଼. On what intervals is 𝑓 increasing?
- 𝑓ᇱ^ ሺ𝑥ሻ ൌ െ sin 𝑥 െ 𝑥 cos 𝑥 for 0 𝑥 𝜋. On which interval(s) is 𝑓 decreasing? 9. 𝑓ᇱ^ ሺ𝑥ሻ ൌ ଵ௫ െ 𝑒 ௫^ sin 𝑥 for 0 ൏ 𝑥 4. On what intervals is 𝑓 decreasing?
For #10-12, calculator use is encouraged.
- The rate of money brought in by a particular mutual fund is represented by 𝑚ሺ𝑡ሻ ൌ ቀ ଶ ቁ
௧ thousand dollars per year where 𝑡 is measured in years. Is the amount of money from this mutual fund increasing or decreasing at time 𝑡 ൌ 5 years? Justify your answer.
- The number of hair follicles on Mr. Sullivan’s scalp is measured by the function ℎሺ𝑡ሻ ൌ 500 𝑒ି ௧^ where 𝑡 is measured in years. Is the amount of hair increasing or decreasing at 𝑡 ൌ 7 years? Justify your answer.
- The rate at which rainwater flows into a street gutter is modeled by the function 𝐺ሺ𝑡ሻ ൌ 10 sin ቀ ௧^
మ ଷ ቁ^ cubic feet per hour where 𝑡 is measured in hours and 0 𝑡 8. The gutter’s drainage system allows water to flow out of the gutter at a rate modeled by 𝐷ሺ𝑡ሻ ൌ െ0.02𝑥 ଷ^ 0.05𝑥 ଶ^ 0.87𝑥 for 0 𝑡 8. Is the amount of water in the gutter increasing or decreasing at time 𝑡 ൌ 4 hours? Give a reason for your answer.
The table above gives values of a function 𝑓 at selected values of 𝑥. If 𝑓 is twice-differentiable on the interval 1 𝑥 5 , which of the following statements could be true?
(A) 𝑓 ᇱ^ is negative and decreasing for 1 𝑥 5.
(B) 𝑓 ᇱ^ is negative and increasing for 1 𝑥 5.
(C) 𝑓 ᇱ^ is positive and decreasing for 1 𝑥 5.
(D) 𝑓 ᇱ^ is positive and increasing for 1 𝑥 5.
5.3 Increasing and Decreasing Intervals^ Test Prep
