Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
Printed by: Guest user
Date: Tuesday, July 23, 2024, 4:53 AM

Description

Work through the odd-numbered problems 1-9. Once you have completed the problem set, check your answers for the odd-numbered questions.

Table of contents

Problems

1. What is the slope of the line through (3,9) and (x, y) for y=x^{2} and x=2.97? x=3.001? \mathrm{x}=3+\mathrm{h}? What happens to this last slope when \mathrm{h} is very small (close to 0)? Sketch the graph of y=x^{2} for x near 3.

3. What is the slope of the line through (2,4) and (x, y) for y=x^{2}+x-2 and x=1.99? \mathrm{x}=2.004? \mathrm{x}=2+\mathrm{h}? What happens to this last slope when \mathrm{h} is very small? Sketch the graph of y=x^{2}+x-2 for x near 2.

5. Fig. 9 shows the temperature during a day in Ames.


(a) What was the average change in temperature from 9 am to 1 pm?

(b) Estimate how fast the temperature was rising at 10 am and at 7 pm?

7. Fig. 11 shows the distance of a car from a measuring position located on the edge of a straight road.

(a) What was the average velocity of the car from \mathrm{t}=0 to \mathrm{t}=20 seconds?
(b) What was the average velocity from \mathrm{t}=10 to \mathrm{t}=30 seconds?
(c) About how fast was the car traveling at \mathrm{t}=10 seconds? at \mathrm{t}=20 \mathrm{~s} ? at \mathrm{t}=30 \mathrm{~s}?

Problem 9 defines new functions A(x) in terms of AREAS bounded by the functions y=3 and y=x+1. This may seem a strange way to define a functions A(x), but this idea will become important later in calculus. We are just getting an early start here.

9. Define A(x) to be the area bounded by the x and y axes, the horizontal line \mathrm{y}=3, and the vertical line at \mathrm{x} (Fig. 13). For example, \mathrm{A}(4)=12 is the area of the 4 by 3 rectangle.

a) Evaluate A(0), A(1), A(2), A(2.5) and A(3).
b) What area would \mathrm{A}(4)-\mathrm{A}(1) represent in the figure?
c) Graph \mathrm{y}=\mathrm{A}(\mathrm{x}) for 0 \leq \mathrm{x} \leq 4.


Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-2.1-Tangent-Lines-Velocities-Growth.pdf

Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 License.


Answers

1. m=\frac{y-9}{x-3}. \quad If x=2.97, then m=\frac{-0.1791}{-0.03}=5.97. If x=3.001, then m=\frac{0.006001}{0.001}=6.001.

If x=3+h, then m=\frac{(3+h)^{2}-9}{(3+h)-3}=\frac{9+6 h+h^{2}-9}{h}=6+h. When h is very small (close to 0), 6+\mathrm{h} is very close to 6.


3. \mathrm{m}=\frac{\mathrm{y}-\mathbf{4}}{\mathrm{x}-2}. \quad If \mathrm{x}=1.99, then \mathrm{m}=\frac{-0.0499}{-0.01}=4.99. If \mathrm{x}=2.004, then \mathrm{m}=\frac{0.020016}{0.004}=5.004.

If x=2+h, then m=\frac{\left\{(2+h)^{2}+(2+h)-2\right\}-4}{(2+h)-2}=\frac{4+4 h+h^{2}+2+h-2-4}{h}=5+h. When \mathrm{h} is very small, 5+\mathrm{h} is very close to 5.


5. All of these answers are approximate. Your answers should be close to these numbers.

(a) average rate of temperature change \approx \frac{80^{\circ}-64^{\circ}}{1 \mathrm{pm}-9 \mathrm{am}}=\frac{16^{\mathrm{o}}}{4 \text { hours }}=4^{\mathrm{o}} per hour.

(b) at 10am, temperature was rising about 5^{\circ} per hour.

at 7pm, temperature was rising about -10^{\circ} per hour (falling about 10^{\circ} per hour).


7. All of these answers are approximate. Your answers should be close to these numbers.

(a) average velocity \approx \frac{300 \mathrm{ft}-0 \mathrm{ft}}{20 \sec -0 \mathrm{sec}}=15 feet per second.

(b) average velocity \approx \frac{100 \mathrm{ft}-200 \mathrm{ft}}{30 \sec -10 \mathrm{sec}}=-5 feet per second.

(c) at \mathrm{t}=10 seconds, velocity \approx 30 feet per second (between 20 and 35 \mathrm{ft} / \mathrm{s}).

at \mathrm{t}=20 seconds, velocity \approx-1 feet per second.

at \mathrm{t}=30 seconds, velocity \approx-40 feet per second.


9. (a) \mathrm{A}(0)=0, \mathrm{~A}(1)=3, \mathrm{~A}(2)=6, \mathrm{~A}(2.5)=7.5, \mathrm{~A}(3)=9.

(b) the area of the rectangle bounded below by the \mathrm{x}-axis, above by the line \mathrm{y}=3, on the left by the vertical line \mathrm{x}=1, and on the right by the vertical line \mathrm{x}=4.

(c) Graph of \mathrm{y}=\mathrm{A}(\mathrm{x})=3 \mathrm{x}.