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Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
Printed by: Guest user
Date: Saturday, May 4, 2024, 4:19 PM

Description

Work through the odd-numbered problems 1-7. Once you have completed the problem set, check your answers.

Table of contents

Problems

Problems 1 – 4  involve estimating slopes of tangent lines.

1) Sketch the lines tangent to the curve shown in Fig. 7 at  x = 1, 2 \; and \; 3.  Estimate the slope of each of the tangent lines you drew.

2) Fig. 8 shows the weight of a "typical" child from age 0 to age 24 months.  (Each of your answers should have the  units  "kilograms per month").

(a) What was the average rate of weight gain from month 0 to month 24?

(b) What was the average weight gain from month 9 to month 12?  from month 12 to month 15?

(c) Approximately how fast was the child gaining weight at age 12 months?  at age 3 months?

3) Fig. 9  shows the temperature of a cup of coffee during a ten minute period. (Each of your answers in (a) – (c) should have the units  "degrees per minute").

(a) What was the average rate of cooling from minute 0 to minute 10?

(b) What was the average rate of cooling from minute 7 to minute 8?  from minute 8 to minute 9?

(c) What was the rate of cooling at minute 8?   at minute 2?

(d) When was the cold milk added to the coffee?

4) Describe a method for determining the slope at the middle of a steep hill on campus

(a) using a ruler, a long piece of string, a glass of water and a loaf of bread.

(b) using a protractor, a piece of string and a helium–filled balloon.

Problems 5 and 6  involve approximating areas.

5) Approximate the area of the leaf in Fig. 4 .

6) Fig. 10 shows temperatures during the month of November.


(a) Approximate the shaded area between the temperature curve and the 65o line from Nov. 15 to Nov. 25.

(b) The area of the "rectangle" is (base)(height) so what are the units of your answer in part (a)?

(c) Approximate the shaded area between the temperature curve and the 65o line from Nov. 5 to Nov. 30.

(d) Who might use or care about these results?

7) Describe a method for determining the volume of a standard incandescent light bulb using a ruler, a tin coffee can, a scale, and a jug of wine.

Answers

1. approx. 1, 0, –1  


 
3.  

(a) Approx.  \dfrac{70 – 150 \; deg.}{10-0 \; min} = 8 \; deg/min. Avg. rate of cooling ≈ 8 \; deg/min. 

(b) Approx. 6 \; deg/min cooling, and  5 \; deg/min cooling. 

(c) Approx. 5.5 \; deg/min cooling, and 10 \; deg/min cooling. 

(d) When t = 6 min.  


 
5. We estimate that the area is approximately (very approximate) 9 cm2.

 
 
7.

Method 1:  Measure the diameter of the coffee can, then fill it about half full of wine and measure the height of the wine and calculate the volume.  Submerge the bulb, measure the height of the wine again, and calculate the new volume.  The volume of the bulb is the difference of the two calculated volumes.

Method 2:  Fill the can completely full of wine and weigh the full can.  Submerge the bulb (displacing a volume of wine equal to the volume of the bulb), remove the bulb, and weigh the can again.  By subtracting, find the weight of the displaced wine and then use the fact that the density of wine is approximately  1 gram per 1  cubic centimeter to determine the volume of the bulb.