Module 5 Review and Practice

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AP Calculus BC Unit 5 Review Sheet I. Differential and Logistic Equations 1. Solve each differential equation: (a)

!" !# !"

= 4đ?‘Ľđ?‘Ś ( , if đ?‘Ś 0 = 2

(b) = đ?‘’ #," !# (c) 1 + đ?‘Ľ ( đ?‘Ś / = 6đ?‘Ś, if đ?‘Ś 0 = 4 1# (d) đ?‘Ś / = "

!"

2. Consider = 3đ?‘Ľ + 2đ?‘Ś. !# (a) Sketch a slope field for the general solution to the given differential equation at the indicated points.

(b) Use Euler’s Method with two steps of equal size to estimate đ?‘Ś(1), if đ?‘Ś 0 = 2 3. Use Euler’s Method with four steps of equal size to approximate the value of đ?‘“(4) if đ?‘Ś = !" đ?‘“(đ?‘Ľ) for = 4 + đ?‘Ľ − 3đ?‘Ś, and if đ?‘“ 0 = 1. !#

!7

4. The rate of growth of a certain population of maniacs is = 0.0003đ?‘ƒ(540 − đ?‘ƒ), where t is !8 time in years. (a) What is the carrying capacity of the population? (b) What will the population be when it is growing most quickly? 5. A population of green gooey monsters grows according to the differential equation !# < = đ?‘Ľ 1 − đ?‘Ľ , where t is measured in days (đ?‘Ą ≼ 0) and x is measured in thousands of green !8 = goody monsters. What is the fastest growth rate that this population exhibits? 6. The rate of change of y is proportional to the value of y. If đ?‘Ś = 4 when đ?‘Ą = 0, and đ?‘Ś = 20 when đ?‘Ą = 10, find the value of the constant of proportionality. *7. A corpse was found in an alley by a detective at 3:15 AM. The detective immediately measured the body temperature and found it to be 89oF. Two hours later, the detective measured the body temperature again and found it to be 72oF. If the air temperature is 45oF, and the assumption is made that the body temperature of the deceased was 98.6oF at the time of death, at what time did the murder occur?

II. Review 8. Write the equation of the tangent line to the curve generated by đ?‘Ľ = < đ?‘Ś = đ?‘Ą ( + 4đ?‘Ą + 3 where đ?‘Ľ = 2.

@

đ?‘Ą + 1 and

(

*9. Calculate the volume of the solid of revolution generated by rotating the region bounded by đ?‘Ś = đ?‘Ľ and đ?‘Ś = đ?‘Ľ ( about the x-axis.

*10. Compute a particle’s speed at đ?‘Ą = 2 if its position is given by < 5đ?‘Ą B − 2đ?‘Ą, đ?‘’ 8,( + 5 >.

11. Write the first four non-zero terms and the general term of the Maclaurin polynomial for đ?‘” đ?‘Ľ = −5cos (3đ?‘Ľ).

*12. Compute the length of the graph of � = � + 4 on the interval 8 ≤ � ≤ 12.

*13. Use a trapezoidal approximation to estimate x 2 4 y 6 5.2

<

= đ?‘” (

đ?‘Ľ đ?‘‘đ?‘Ľ using the information below: 5 8 3.25 2.1

*14. If đ?‘” −2 = 5 and đ?‘”/ −2 = , approximate đ?‘”(−1.95). 1

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