Instructions for Recitation R01, Tue Sep 26, 2025

• Students will work on the problems in the packet called Recitation R01 Problems . (The Recitation Instructor will hand out printed copies at the beginning of the Recitation session.)
• Students should work in groups of two.
  • If there is an extra student, there can be one group of three .
  • There should be no students working alone.
• The Recitation Instructor will hand out one copy of the packet to each group.
• All the students in the group should print their names on the packet.
• Students should complete as much of the packet as they can, and then hand the packet to the Recitation Instructor at the end of the Recitation period.

• Worksheet: Using the Squeeze Theorem
• Worksheet: Using the Intermediate Value Theorem

Eight Problems to Be Done in Recitation R02 for MATH 2301 (Barsamian)

[1] A limit that is an Indeterminate Form and that require no trick, just messy work

(This problem is similar to book exercise 1.4#15, which is not assigned. It is also similar to Book Section 1.4 Example 2)
Find the limit $$\lim_{t\rightarrow -2}\frac{t^2-t-6}{2t^2+5t+2}$$
Show valid steps that lead to your answer.

[2] Another limit that is an Indeterminate Form and that require no trick, just messy work

(This problem similar to homework exercise 1.4#25)
Find the limit $$\lim_{x\rightarrow -5}\frac{\frac{1}{5}+\frac{1}{x}}{5+x}$$
Show valid steps that lead to your answer.

[3] A limit that Involves Rationalizing

(This problem is similar to homework exercise 1.4#21 and Book Section 1.4 Example 5)
Find the limit $$\lim_{h\rightarrow 0}\frac{\sqrt{9+h}-3}{h}$$
Show valid steps that lead to your answer.

[4] A Limit that involves the Absolute Value

(This problem is similar to homework exercise 1.4#38, and similar to Book Section 1.4 Example 7)
Find the limit $$\lim_{x\rightarrow -4}\frac{3x+12}{|x+4|}$$
Show valid steps that lead to your answer.

[5] A Limit involving the Squeeze Theorem

(This problem is similar to homework exercise 1.4#35 and Book Section 1.4 Example 9.)
Prove that that $$\lim_{x\rightarrow 0}\left[x^2\cos{\left(\frac{3}{x}\right)}\right]=0$$
Show valid steps that lead to your answer.
Hint: Use the Worksheet entitled Using the Squeeze Theorem to organize your work.

[6] A limit that uses the famous fact that \(\lim_{x\rightarrow 0}\frac{\sin{(x)}}{x}=1\)

(This problem is similar to homework exercises 1.4#51 and is related to book Section 1.4 Example 10)
Find the limit $$\lim_{5\rightarrow 0}\frac{\tan{(12t)}}{\sin{(3t)}}$$ Show valid steps that lead to your answer.
Warning: Don�t be tempted to replace \(\tan{(12t)}\) with \(12\tan{(t)}\) because \(\tan{(12t)} \neq 12\tan{(t)}\)
Hint: Replace \(tan{(12t)}\) with \(\frac{\sin{(12t)}}{\cos{(12t)}}\)

[7] Problem involving the Intermediate Value Theorem

(a) Let \(f(x)=x+\sqrt[3]{x}-1\). Use the Intermediate Value Theorem to show that \(f(x)\) has a root on the interval \((0,1)\). That is, show that there exists an \(x\) value, with \(0 \lt x \lt 1\), such that \(f(x)=0\). Explain clearly.
Hint: Use the Worksheet entitled Using the Intermediate Value Theorem to organize your work.

(b) (This problem is book exercise 1.5#40, which is similar to homework exercise 1.5#39) Use the Intermediate Value Theorem to show that there is a solution of the equation $$\sqrt[3]{x}=1-x$$ on the interval (0,1). Explain clearly.

[8] Problem involving the Intermediate Value Theorem

(a) Let \(f(x)=\cos{(x)}-x\), where \(x\) is in radians , not degrees . Use the Intermediate Value Theorem to show that \(f(x)\) has a root. That is, show that there exists an \(x\) value such that \(f(x)=0\). Explain clearly.
Hint: Use the Worksheet entitled Using the Intermediate Value Theorem to organize your work.

(b) (This problem is similar to homework exercise 1.5#43) Use the Intermediate Value Theorem to show that there is a solution of the equation $$\cos{(x)}=x$$ where \(x\) is in radians , not degrees . Explain clearly.

Basic Derivative Formulas

Derivative of a Constant Function If \(c\) is a constant, then $$\frac{d}{dx}(c)=0$$

The Power Rule If \(n\) is any real number, then $$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

The Sum Constant Multiple Rule If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then $$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

The Sine and Cosine Rules $$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$ $$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$

[1] (2.2#23) For the function $$g(x)=\frac{1}{x}$$

1. Find \(g'(x)\) using the Definition of the Derivative $$g'(x)=\lim_{h\rightarrow 0} \frac{g(x+h)-g(x)}{x}$$ That is, build the limit and find its value. ( Do not use the Derivative Rules !!) Show all steps clearly and explain key steps.
2. Start over. Find \(g'(t)\) again, but this time use the Derivative Rules . (You should get the same result that you got in (a).

[2] (2.3#2) Use the Derivative Rules to find the derivative of the function $$f(x) = \pi^2$$ Show all details clearly and use correct notation.

[3] (2.3#7)

• NO CALCULATORS!!
• Give exact answers in symbols, not decimal approximations.
• You may have to review the \((x,y)\) coordinates of the points where the rays of the famous angles intersect the Unit Circle .

1. Find \(f(\pi)\).
2. Find \(f'(x)\) using the Derivative Rules
3. Find \(f'(\pi)\)
4. Find the height of the graph of \(f(x)\) at \(x=\pi\).
5. Find the slope of the graph of \(f(x)\) at \(x=\pi\).

[4] (2.3#19) For the function $$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$

1. Rewrite \(f(x)\) in power function form . That is, write it in the form $$f(x)=ax^p+bx^q+cx^r$$ where \(a,b,c,p,q,r\) are real numbers.
2. Find \(f'(x)\) using the Derivative Rules

[5] (2.3#21) For the function $$v=t^2-\frac{1}{\sqrt[4]{t^3}}$$

1. Rewrite the function in power function form . That is, write it in the form $$v(t)=at^p+bt^q$$ where \(a,b,p,q\) are real numbers.
2. Find \(v'(t)\) using the Derivative Rules

Review of Tangent Lines And Normal Lines

Remember that the line tangent to the graph of \(f(x)\) at \(x=a\) is the line that has these two properties

• The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\), called the point of tangency
• The line has slope \(m=f'(a)\)

A new thing, the line normal to the graph of \(f(x)\) at \(x=a\) , is the line that has these two properties

• The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\)
• The line is perpendicular to the line that is tangent to the graph at that point. That is,
  • If the tangent line has slope \(m_T\neq 0\), then the normal line has slope $$m_N=-\frac{1}{m_T}$$
  • If the tangent line has slope \(m_T = 0\), which indicates that the tangent is horizontal , then the normal line is vertical .

[6] (2.3#27)

• NO CALCULATORS!!
• Give exact answers in symbols, not decimal approximations.
• You may have to review the \((x,y)\) coordinates of the points where the rays of the famous angles intersect the Unit Circle .

1. Find the equation for the line tangent to the graph of \(f(x)\) at \(x=\frac{2\pi}{3}\) .
2. Find the equation for the line normal to the graph of \(f(x)\) at \(x=\frac{2\pi}{3}\) .
3. Draw the graph and draw your tangent line and normal line. Label important stuff.

Details TBA