Mathematics 265 Introduction to Calculus I

Study Guide :: Appendix B

Learning from Mistakes—Hints

Unit 1

1. If ${\displaystyle x}$ is any number in the interval ${\displaystyle \left[4,\infty \right)\text{,}}$ in which interval is ${\displaystyle 1-\frac{x}{2}\text{?}}$

   Hint: See “Rules for Inequalities” (page 338 of the textbook).

2. Simplify each of the expressions below.

   1. ${\displaystyle {\left(a+3b\right)}^{2}a{b}^{-1}}$
   2. ${\displaystyle {(x^3{y}^{-3})}^2}$
   3. ${\displaystyle (a{b}^2-b{c}^3)abc}$

   Hint: See the laws of exponents on the “Reference Pages” at the beginning of the textbook.

3. Factor each of the expressions below.

   1. ${\displaystyle 2x^2-7x-4}$
   2. ${\displaystyle 3x^5+24x^2}$
   3. ${\displaystyle 81x^{4}y-y^5}$

   Hint: See “Factoring Special Polynomials” on the “Reference Pages” at the beginning of your textbook.

4. 1. Simplify the expression ${\displaystyle \frac{x^2+9x+20}{x^2+5x}\text{.}}$
   2. Rationalize the expression ${\displaystyle \frac{\sqrt{4+h}-2}{h}\text{.}}$

   Hint: Review factorization and rationalization.

5. Give the exact values of each of the trigonometric functions below.

   1. ${\displaystyle \sec \left(\frac{7\pi }{6}\right)}$
   2. ${\displaystyle \sin \left(\frac{7\pi }{12}\right)}$

   Hint: See the table of exact values on page 358 of the textbook, and locate in the unit circle the angles given.

Unit 2

1. Define each of the functions below as a set of pairs.
   1. The velocity ${\displaystyle v}$ depends on the time ${\displaystyle t\text{.}}$
   2. The bacteria population ${\displaystyle B}$ depends on the amount of oxygen ${\displaystyle o\text{.}}$

   Hint: See Definition 2.3.

2. Let ${\displaystyle S(t)=3.75t^2+500}$ be a function where ${\displaystyle S}$ is the salary for the number of units sold ${\displaystyle t\text{.}}$ Find the value of ${\displaystyle S}$ at ${\displaystyle t=4-\sqrt{14}\text{.}}$

   Hint: Review Unit 1.

3. Find the domain of each of the functions below.
   1. ${\displaystyle g(t)=\frac{4-t}{6-\sqrt{t^2-9}}}$
   2. ${\displaystyle h(x)=\tan (2x)}$

   Hint: See Definition 2.6.

4. Sketch the graph of a single function that satisfies all of the conditions listed below.
   • Domain is [3, 12].
   • The pairs (4, 5) and (5, 4) are not in the function.
   • The function is constant equal to 1 on the interval (8, 12].

   Hint: See the section on the graph of a function in Unit 2.

5. If the graph of the function ${\displaystyle f}$ is shown below, give a sketch of each of the following functions:

   

   1. ${\displaystyle F(x)=2f(x)-1.}$
   2. ${\displaystyle F(x)=f(x+2)+3.}$

   Hint: Review the section titled “Transformations of Functions” on pages 30-34 of the textbook.

Unit 3

1. Draw the graph of a single function ${\displaystyle f}$ such that
   1. ${\displaystyle \underset{x\to 0}{\lim }f(x)=3}$
   2. ${\displaystyle \underset{x\to -2}{\lim }f(x)=\infty }$
   3. ${\displaystyle \underset{x\to \infty }{\lim }f(x)=-1.}$

   Hint: See the vertical line test.

2. The graph of a function ${\displaystyle f}$ is shown below. Give the following limits:

   

   1. ${\displaystyle \underset{x\to -4^{-}}{\lim }f(x)}$
   2. ${\displaystyle \underset{x\to 2}{\lim }f(x)}$
   3. ${\displaystyle \underset{x\to \infty }{\lim }f(x)}$
   4. ${\displaystyle \underset{x\to -4^{+}}{\lim }f(x)\text{.}}$

   Hint: See the section on visual evaluation of limits in Unit 3.

3. Evaluate each of the limits below. If a limit does not exist, explain why.

   1. ${\displaystyle \underset{x\to 0^{+}}{\lim }x\sin x+\frac{1}{x}}$

   2. ${\displaystyle \underset{x\to 1}{\lim }\frac{x^2+2x-15}{x^2+x-12}}$

   3. ${\displaystyle \underset{x\to 0}{\lim }\frac{x^2-1}{x^2+x^4}}$

   4. ${\displaystyle \underset{x\to \pi ∕2^{-}}{\lim }\text{}\tan x+\sec x}$

   5. ${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{\tan (3x)}{\sin (4x)}}$

   6. ${\displaystyle \underset{x\to 3}{\lim }\frac{\sin (x-3)}{x^2+x-12}}$

   7. ${\displaystyle \underset{x\to \infty }{\lim }x\cos \left(\frac{1}{x}\right)}$

   8. ${\displaystyle \underset{x\to \infty }{\lim }\frac{\cos x}{x}}$

   9. ${\displaystyle \underset{x\to \infty }{\lim }\frac{x^2+3}{\sqrt{x^2+x-1}}}$

   10. ${\displaystyle \underset{x\to -\infty }{\lim }\frac{\sqrt{\left(3x+2\right)\left(x+1\right)}}{3-x}}$

   Hint: Study the examples and warnings in this unit. Each step in the evaluation of a limit must be justified by the application of a definition, theorem, proposition or corollary.

4. Find the vertical and horizontal asymptotes of the function

   ${\displaystyle g(x)=\frac{\sqrt{x^4+4x^3+4x^2}}{3x^2+5x-2}\text{.}}$

   Hint: See Definition 6 on page 56 of the textbook.

Unit 4

1. The displacement (in km) of a moving car is given by ${\displaystyle s(t)=3t^3+4t-2\text{,}}$ where ${\displaystyle t}$ is measured in hours.
   1. Give the average velocity in the time period ${\displaystyle \left[1,5\right]\text{.}}$
   2. Give the two different interpretations of value found in part (a).
   3. Give the average velocity in the time period ${\displaystyle \left[1,1+h\right]}$ for ${\displaystyle h>0\text{.}}$

   Hint: See the section on the average rate of change and the slope of the secant line in Unit 4.

2. Use Definition 4.4 to find ${\displaystyle \frac{d}{dx}\frac{3}{x}\text{.}}$

   Hint: See Example 4.15.

3. Consider the piecewise function

   ${\displaystyle f(x)=\{\begin{array}{ll}\left|x\right|\hfill & \text{for }x\le 4\hfill \\ {(x-6)}^2\hfill & \text{for }x>4\hfill \end{array}}$

   1. Sketch the graph of the function ${\displaystyle f\text{.}}$
   2. Sketch the graph of the derivative function ${\displaystyle f^{\prime }\text{.}}$

   Hint: Sketch each piece independently, and then sketch them together at the point where ${\displaystyle x=4\text{.}}$

4. Give the indicated derivatives.
   1. ${\displaystyle f(x)=\frac{3x^2+6x-3}{x-4x^3}{f}^{\prime }(x)\text{.}}$
   2. ${\displaystyle \frac{d}{dx}\sqrt{x^3\sin x}\text{.}}$
   3. ${\displaystyle \frac{d}{dx}\sec \sqrt{x+1}\text{.}}$
   4. ${\displaystyle \frac{d^2}{d{x}^2}\tan \left(3x\right)\text{.}}$

   Hint: Review the rules of differentiation.

5. A particle is moving along the curve ${\displaystyle y=\sqrt{x}\text{.}}$ As the particle passes through the point ${\displaystyle \left(4,2\right)\text{,}}$ its ${\displaystyle x}$-coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?

   Hint: Check the section titled “Implicit Differentiation,” pages 182-186 of the textbook.

6. A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

   Hint: Check the section titled “Implicit Differentiation,” pages 182-186 of the textbook.

Unit 5

1. Sketch the graph of the function ${\displaystyle f(x)={\left(x-1\right)}^{3∕2}\text{.}}$

   Hint: See the section titled “Guidelines for Sketching a Curve” on pages 232-236 of the textbook.

2. Find the local extreme points of the function ${\displaystyle g(x)=x^6+6x^4\text{.}}$

   Hint: Read “The Second Derivative Test,” on page 215 of the textbook.

3. Find the extreme values of the function ${\displaystyle h(x)=x^3-6x}$ on the interval ${\displaystyle \left[0,2\right]\text{.}}$

   Hint: Read “The Closed Interval Method” on page 225 of the textbook.

4. Find a number greater or equal to 2 such that the sum of the number and its reciprocal is as small as possible.

   Hint: Reread the problem carefully.

Unit 6

1. Integrate

   ${\displaystyle \int {\tan }^2\left(3x\right){\sec }^2\left(3x\right)dx.}$

   Hint: See the section on antiderivatives and the general power rule in Unit 6.

2. Solve ${\displaystyle {\int }_1^5{x}^3\sqrt{x^2+1}dx\text{.}}$

   Hint: See Example 6.34.

3. Solve ${\displaystyle {\int }_0^1{\left(x^2-\sqrt{x}\right)}^{2}dx\text{.}}$

   Hint: See the Fundamental Theorem of Calculus, part 2 in Unit 6.

4. Find a function ${\displaystyle f(x)}$ that satisfies the following two conditions:
   • ${\displaystyle f^{\prime }(x)=\cos x+\sqrt{3x}}$
   • ${\displaystyle f(1)=1\text{.}}$

   Hint: See Example 6.5.

5. Let ${\displaystyle f(x)=4x^3-6x^2+3}$ be a continuous function. Find the value ${\displaystyle c}$ in the interval ${\displaystyle \left[-1,0\right]}$ which satisfies the Mean Value Theorem.

   Hint: See the Mean Value Theorem in Unit 6.

6. Find the area below the curve of the function ${\displaystyle f(x)=x^3+4x}$ on the interval ${\displaystyle \left[-1,3\right]\text{.}}$

   Hint: See Example 6.31.

7. Find the derivative of the function ${\displaystyle g(x)={\int }_{3x}^{4}t\tan tdt\text{.}}$

   Hint: See Example 6.37.

Unit 7

1. A spacecraft uses a sail and the “solar wind” to produce a constant acceleration of 0.032 m/s². Assuming that the spacecraft has a velocity of 60 km/h when the sail is first raised, how far will the spacecraft travel in 1 hour, and what will its velocity be at the end of this hour?

   Hint: Check the units.

2. Find the area under the curve ${\displaystyle y=x^2+x-2}$ on the interval ${\displaystyle \left[0,3\right]\text{.}}$

   Hint: An area under a curve must be positive.

3. Find the area between the curves ${\displaystyle y=\frac{1}{x^2}\text{,}}$ ${\displaystyle y=x}$ and ${\displaystyle y=\frac{x}{8}\text{.}}$

   Hint: Sketch a graph of the region bounded by these curves.

4. A spring exerts a force of 100 N when it is stretched 0.2 m beyond its natural length. How much work is required to stretch the spring 0.8 m beyond its natural length?

   Hint: Review Hooke’s Law on page 327 of the textbook.