Chapter 6 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem · clarified

   You open up a business using $10 000 of your hard-earned money. You think that the rate of return on your investment (as a decimal) can be modeled by the equation $r=e^{-t}+0.07$. What is the rate of increase of the assets in your business after 3 years? Use units of dollars per year.

   Clarification: Use the asset model $A\left(t\right)=10000e^{t(e^{-t}+0.07)}$. Here the given r(t) is inserted into the exponent t r(t); it is not an instantaneous proportional growth rate.

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2. Original problem · clarified

   You need $50 000 per year to live comfortably when you retire. Your bank offers a high interest savings account that gives you a compounding interest of 5% per year. How much money do you need to invest now if you want to live comfortably off the interest during your retirement 30 years from now ?

   Clarification: Assume continuous compounding. The retirement fund is chosen so that interest at 5% supplies $50,000 per year without using the principal.

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3. Original problem

   You want to maximize revenues for your company by changing the price. Your sales follow the function $S=8-3\ln P$, where S is the quantities sold (in 10 000 units) and P is the price. You want to keep the price of your product between $1 and $12. ​
   a) What is the optimum price point?​
   b) How many units will you sell at this price?

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4. Original problem

   You want to maximize profits by changing the quantity of laptops your company produces. Each laptop brings in $1000 of revenue. The total cost of producing laptops per year is represented by $C=\frac{Q^3}{\mathrm{10,000}}$, where C is the cost and Q is the quantity of laptops produced.​
   a) What is the optimum number of laptops your company should make for profit maximization?​
   b) What is the annual profit at that quantity?

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5. Original problem

   As the size of your company increases, its percentage growth decreases. However, your company is adding more employees per year than ever before! Explain these concepts using variables, derivatives, and related rates.

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6. Original problem

   What is the absolute maximum value of the graph of $y=10x-x^2$

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7. Original problem

   For the equation $y=3\ln x-\sqrt{x}$, ​
   a) At which point does its absolute maximum value occur?​
   b) What is the maximum value?

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8. Original problem

   As you pump air into a spherical balloon at a rate of 3cm³ per second, its radius increases to 2 cm. What is the rate of change of its surface area?

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9. Original problem

   You have an inflatable cube that constantly increases in size at 1cm per second. What is the rate of change in its volume when one of its sides is 4 cm?

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10. Original problem · clarified

    Your friend gets distracted while holding up your ladder and you feel yourself starting to fall. Assuming gravity causes the vertical displacement from your original position to be defined as $x=-3t^2$, at what rate is the angle between the ladder and the ground changing after 0.5s?​
    

    Clarification: Treat the ladder as rigid, with its foot sliding on horizontal ground. Use the source diagram’s initial height 5 m and base distance 3 m; the top height is 5−3t² metres.

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11. Added problem

    Find and classify the local extrema of $f\left(x\right)=x^3-3x$.

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12. Added problem

    Find the absolute extrema of $f\left(x\right)=x^2+2x-1$ on $[-2,2]$.

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13. Added problem

    A student says every point with f′=0 is a maximum or minimum. Use $f\left(x\right)=x^3$ to test the claim.

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14. Added problem

    A rectangular garden uses 40 m of fence on all four sides. Find the largest possible area and dimensions.

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15. Added problem

    A river forms one side of a rectangular garden. With 40 m of fence for the other three sides, find the dimensions maximizing area.

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16. Added problem

    Cut equal squares from the corners of a 12 cm square sheet and fold an open box. Find the cut size maximizing volume.

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17. Added problem

    A closed cylindrical can holds $16\pi$ cm³. Find the radius and height minimizing its surface area.

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18. Added problem

    Find the point on $y=x$ closest to $(2,0)$.

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19. Added problem

    A fictional shop sells $q=200-10p$ units at price p dollars. On $0\le p\le 20$, find the price maximizing revenue.

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20. Added problem

    A circular ripple has radius 3 cm growing at 2 cm/s. How fast is its area changing?

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21. Added problem

    A sphere gains volume at $8\pi$ cm³/s when its radius is 2 cm. Find its radius rate.

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22. Added problem

    A 5 m ladder rests on level ground against a wall. Its base moves away at 2 m/s. How fast does the top fall when the base is 3 m from the wall?

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23. Added problem

    A cone-shaped tank has height 6 m and top radius 3 m, point downward. Water enters at 1 m³/min. Find the height rate when the water is 2 m deep.

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24. Added problem

    Two walkers leave the same point at the same time. One goes east at 3 m/s and one north at 4 m/s. How fast does their separation grow?

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25. Added problem

    A rectangle is 5 cm long and 4 cm wide. Its length shrinks at 1 cm/s while its width grows at 2 cm/s. Find its area rate.

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26. Added problem

    A sign chart gives f′>0 on $(-\infty ,-2)$, f′<0 on $(-2,1)$, and f′>0 on $(1,\infty )$. Assume f is continuous and stationary at −2 and 1. Classify the two points.

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27. Added problem

    Does $f\left(x\right)=x$ have absolute extrema on $(0,1)$? Explain the role of the endpoints.

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28. Added problem

    Maximize $f\left(x\right)=20-(x-5{)}^2$ on $[0,3]$.

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29. Added problem

    A fictional team has 400 employees and gains 12 employees/year at the current instant. State its absolute and percentage growth rates.

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30. Added problem

    Two positive numbers have product 36. Find the pair with the smallest sum.

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