MATH 2301-100 Spring 2022 Calculus I Test Guide

PreCalculus

Fractions and exponents

1. Determine whether each of the following statements is True or False.
   1. \(\frac{A+B}{A+C}=\frac{B}{C}\)
   2. \(\frac{A+B}{C+D}=\frac{A}{C}+\frac{B}{D}\)
   3. \(x+x^{-1}=0\)
   4. \((x+y)^2 = y^2+x^2+2xy\)
   5. \(x^3 + y^3 = (x+y)(x^2 + xy + y^2)\)
2. Use the properties of exponents to simplify \[ \left(\frac{25}{4x^4y^5}\right)\left(\frac{5}{2x^3y^2}\right)^{-3}\,.\]

Lines

1. Write the equation of the line passing through the two points \((1,3)\) and \((3,4)\).
2. Consider \(y=f(x)\) with graph:
   
   1. Find the equations of the lines that comprise the graph of \(f\).
   2. Draw the graph of \(g(x)=2f(-x)+1\). Mark and label the points corresponding to \(A\), \(B\), and \(C\).

Trigonometry

1. Determine whether each of the following statements is True or False.
   1. \(\frac{\sin(x^3)}{\sin(x^2)}=\frac{x^3}{x^2}=x\)
   2. \(\sin(x+y)=\sin(x)+\sin(y)\)
2. Verify the identity \(\frac{1}{1-\cos(\theta)}+\frac{1}{1+\cos(\theta)} = 2 \csc^2(\theta)\).

Exponentials and Logarithms

1. Solve the following equation for \(x\): \(\log_3(x-4)+\log_3(x+4) = 2\).
2. Use the properties of logarithms to write \(f(x) = 2 \ln(x - 3) + \log_e (y + 2) - \ln(z)\) as a single logarithm.

Function operations

1. Determine whether each of the following statements is True or False.
   1. \(\sqrt{x^2+y^2}=\sqrt{x^2}+\sqrt{y^2}=x+y\)
   2. \(f(x)=x^{-1}\;\Rightarrow\;f(x+h)=x^{-1}+h\)
2. With \(f(x)=\tan x\), \(g(x)=\frac{x}{x-1}\), and \(h(x)=\sqrt[3]{x}\), compute \((f\circ g\circ h)(x)\).

Simplification methods

1. Simplify and cancel so that you can plug in the given value without dividing by 0. Then plug in the value.
   1. For \(x=2\), \( \frac{x^2+x-6}{x-2}=\)
   2. For \(x=4\), \( \frac{\sqrt{x}-2}{x-4}=\)
   3. For \(h=0\), \( \frac{(x+h)^2 -x^2}{h}=\)
   4. For \(h=0\), \( \frac{(x+h)^{-1} - x^{-1}}{h}=\)
   5. For \(h=0\) and \(f(x)=2x^2\), \(\frac{f(x+h) -f(x)}{h}=\)

1.1 An Introduction To Limits

Exercise 4

1.2 Epsilon-Delta Definition of a Limit

Exercise 5

1. Define "\(\displaystyle \lim_{x\rightarrow c} f(x) =L\)".

1.3 Finding Limits Analytically

Exercises 7-18, 27-33, 35-38, 39

1. Compute the following limits:
   1. \(\displaystyle\lim_{x\rightarrow 2} \frac{x^2+x-6}{x-2}=\)
   2. \(\displaystyle\lim_{x\rightarrow 4} \frac{\sqrt{x}-2}{x-4}=\)
   3. \(\displaystyle\lim_{h\rightarrow 0} \frac{(x+h)^2 -x^2}{h}=\)
   4. \(\displaystyle\lim_{h\rightarrow 0} \frac{(x+h)^{-1} - x^{-1}}{h}=\)
   5. For \(f(x)=2x^2\), \(\displaystyle\lim_{h\rightarrow 0}\frac{f(x+h) -f(x)}{h}=\)
2. State the Squeeze Theorem using the template below.
   • If ...
   • then ...
3. Use the Squeeze Theorem to evaluate \[\lim_{x\rightarrow 0} \left(x^2 \cos\left(\frac{1}{x}\right)+1\right)\,.\]
4. Compute the following limits. If you use the squeeze theorem, then indicate the two functions that you are using to squeeze. (Do not use L'Hopital's rule.)
   1. \(\displaystyle\lim_{x\rightarrow 2}\frac{x-2}{x^2-5x+6}\)
   2. \(\displaystyle\lim_{x\rightarrow 1}\frac{\sqrt{x}-1}{x-1}\)
   3. \(\displaystyle\lim_{x\rightarrow 0}x^2 \cos(3/x)\)
   4. \(\displaystyle\lim_{h\rightarrow 0}\frac{x^2-(x-2h)^2}{h}\)
   5. \(\displaystyle\lim_{t\rightarrow 0}\frac{\frac{1}{5+t}-\frac{1}{5}}{t}\)
   6. For \(f(x)= (2x+1)^{-1}\), compute \( \displaystyle\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\)

1.4 One-Sided Limits

Exercises 5-21

1. Sketch the graph of a single function \(f\) that:
   1. has domain \([-4,5]\)
   2. has \(f(2) = 1\)
   3. has \(\displaystyle \lim_{x\rightarrow 2}f(x)=4\)
   4. has \(\displaystyle \lim_{x\rightarrow 3^+}f(x)=-3\)
   5. has \(\displaystyle \lim_{x\rightarrow 3^-}f(x)=3\)

Material from previous Sections

1.5 Continuity

Exercises 1-10

1. Sketch the graph of a single function \(f\) that:
   1. has domain \([-4,5]\)
   2. has \(f(2) = 1\)
   3. has \(\displaystyle \lim_{x\rightarrow 2}f(x)=4\)
   4. has \(\displaystyle \lim_{x\rightarrow 3^+}f(x)=-3\)
   5. has \(\displaystyle \lim_{x\rightarrow 3^-}f(x)=3\)
   6. is continuous except possibly at \(x=1\), \(x=2\), and \(x=3\)
2. Consider the function \[f(x)= \begin{cases} x^2 & \text{if $x\le -2$}\\ Ax & \text{if $x> -2$} \end{cases}\, , \] where \(A\) is some constant.
   1. Find \(\displaystyle\lim_{x\rightarrow -2^-}f(x)\). Is \(f\) continuous from the left at \(x=-2\)?
   2. What value of \(A\) would make \(f\) continuous at \(x=-2\)?
   3. Using the value of \(A\) that you just found, graph \(f\).
3. Let \[ f(x)= \begin{cases} {\displaystyle\frac{2x^2-x-15}{x-3}} & \text{if \(x \lt 3\)}\\ kx-1 & \text{if \(x \ge 3\)} \end{cases}\, . \] Determine the value of \(k\) that will make the function \(f\) continous at 3, or explain why no value of \(k\) will work.
4. State the Intermediate Value Theorem using the template below.
   • If ...
   • then ...
5. Use the Intermediate Value Theorem to show that the equation \(x^2 =\cos(x)\) has a solution.

1.6 Limits Involving Infinity

Exercises 4-7, 19-28

1. Sketch the graph of a function \(f\) for which:
   1. It has a removable discontinuity at \(x=1\),
   2. \(\lim_{x\rightarrow 2^-}f(x)=+\infty\),
   3. \(\lim_{x\rightarrow 2^+}f(x)=-1\),
   4. \(\lim_{x\rightarrow +\infty}f(x)=3\),

2.1 Instantaneous Rates of Change: The Derivative

Exercises 10, 11, 13, 14, 18, 19, 21, 22, 29-32

1. State the definition of the derivative as a limit.
2. Use the definition of the derivative as a limit to compute the derviative of \(f(x)=\sqrt{x}+3\).
3. Complete the definitions:
   1. Definition: A function \(f\) is continuous at \(x=a\) if ...
   2. Definition: A function \(f\) is differentiable at \(x=a\) if ...
   Give an example of a function that is one but not the other.
4. Sketch the graph of a differentiable function \(g\) for which \(g(0)=g'(0)=0\), \(g'(-1)=-1\), \(g'(1)=3\), and \(g'(2)=1\).
5. Determine whether each of the following statements is True or False.
   1. If \(f'(a)\) exists then \(\displaystyle\lim_{t\rightarrow a}f(t)=f(a)\).
   2. If \(f\) is continuous at \(a\) then \(\displaystyle\lim_{t\rightarrow a^+}f(t)=f(a)\).
   3. If the velocity of an object is constant, then its acceleration is zero.
   4. The function \(f(x)=x^{2/3}\) is differentiable on \((-\infty,\infty)\).

2.2 Interpretations of the Derivative

Exercises 5

2.3 Basic Differentiation Rules

Exercises 11-25, 27-38

2.4 The Product and Quotient Rules

Exercises 15-48

2.5 The Chain Rule

Exercises 7-40

Material from previous Sections

2.6 Implicit Differentiation

Exercises 5-25, 27-32, 37-40

1. Use implicit differentiation to find an equation for the tangent line to the curve defined by \( \displaystyle x^2+4xy+y^2=13 \) at the point \((1,2)\).

2.7 Derivatives of Inverse Functions

Exercises 5-24, 28, 29

1. Determine whether each of the following statements is True or False.
   1. If \(f\) is a one-to-one continuous function, then \(f^{-1}\) is a one-to-one continuous function.
   2. If \(f\) is a one-to-one differentiable function, then \(f^{-1}\) is a one-to-one continuous function.

3.1 Extreme Values

Exercises 17-26

3.2 The Mean Value Theorem

Exercises 3-20

1. 1. State Rolle's Theorem using the template....
   2. Verify that the function \(f(x)=\dots\) satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem.
2. 1. State the Mean Value Theorem (MVT) using the template....
   2. Verify that the function \(f(x)=\dots\) satisfies each of the hypotheses of the MVT on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the MVT.
3. If \(f\) is a differentiable function with \(f(0)=2\) and \(f'(x) \le 4\), then what is the maximum value \(f(3)\) could be? What is the minimum value \(f(3)\) could be?
4. Let \(f\) be a continuous function with \(f(0)=3\), \(f(2)=6\), \(f'(x)=0\) for \(0 < x < 1\), and \(f'(x) < 2\) for \(1 < x < 2\). Sketch such a function or explain why it is impossible. Hint Break into intervals and use the Mean Value Theorem.

3.3 Increasing and Decreasing Functions

Exercises 3, 5, 15-24

3.4 Concavity and the Second Derivative

Exercises 1-4, 15-56

3.5 Curve Sketching

Exercises 6-28

1. For each part below, sketch the graph of a function that satisfies all the conditions or explain why it is impossible to satisfy all the conditions. Conditions given in terms of \(x\) (such as \(f(x) > 0\)) apply for all \(x\), whereas conditions given in terms of \(a\) (such as \(f'(a) > 0\)) apply for a single value \(x=a\), which you should label.
   1. \(f(x) > 0\), \(f'(x) > 0\), and \(f''(x) > 0\)
   2. \(f(x) < 0\), \(f'(x) > 0\), and \(f''(x) > 0\)
   3. \(f(x) > 0\), \(f'(x) < 0\), and \(f''(x) > 0\)
   4. \(f(x) > 0\), \(f'(x) > 0\), and \(f''(x) < 0\)
   5. \(f(x) > 0\), \(f'(x) < 0\), and \(f''(x) < 0\)
   6. \(f(x) < 0\), \(f'(x) > 0\), and \(f''(x) < 0\)
   7. \(f(x) < 0\), \(f'(x) < 0\), and \(f''(x) > 0\)
   8. \(f(x) < 0\), \(f'(x) < 0\), and \(f''(x) < 0\)
   9. \(f'(a) = 0\), and \(f''(x) > 0\)
   10. \(f'(a) = 0\), and \(f''(x) < 0\)
   11. \(f(a) = 3\), \(f'(x) > 0\), and \(f''(x) = 0\)
   12. \(\displaystyle \lim_{x\rightarrow a^-}f(x)=-2\), \(\displaystyle \lim_{x\rightarrow a^+}f(x)=1\), \(f(a)=3\)
   13. \(\displaystyle \lim_{x\rightarrow a}f(x)=4\), \(f(a)=3\), \(f'(a)=0\)
   14. \(f(a)=1\), \(f(a+1)=2\), \(f'(x) > 0\)
   15. \(f(a)=1\), \(f(a-1)=2\), \(f'(x) > 0\)
2. Sketch the graph of a single function that has all of the following properties:
   1. ....
   2. ....
3. For \(f(x)=...\)
   1. Find the intervals where \(f\) is increasing, and the intervals where it is decreasing.
   2. Find the local maximum and minimum values
   3. Find the intervals where \(f\) is concave up, and the intervals where it is concave down.
   4. Find the inflection points
   5. Use this information to sketch the graph.
4. For \(f(x)=...\)
   1. Find the domain.
   2. Find the intercepts.
   3. Determine any symmetries.
   4. Find any asymptotes.
   5. Find the intervals on which \(f\) is increasing or decreasing.
   6. Find the local maximum and minimum values of \(f\).
   7. Find the intervals of concavity and the inflection points.
   8. Use the information above to sketch the graph.
5. Determine whether each of the following statements is True or False.
   1. A local maximum of a function \(f(x)\) can only occur at a point where \(f'(x)=0\).
   2. If \(f\) is a continous function on the interval \([a,b]\) then \(f\) attains an absolute maximum at some number \(c\) in \((a,b)\).
   3. If \(f''(a)=0\) then \(f\) has an inflection point at \(x=a\).
   4. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(f+g\) is an increasing function on \(I\).
   5. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(f-g\) is an increasing function on \(I\).
   6. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(fg\) is an increasing function on \(I\).
   7. If \(f\) and \(g\) are positive increasing functions on an interval \(I\) then \(fg\) is an increasing function on \(I\).

Material from previous Sections

4.1 Newton's Method

Exercises 5, 6, 8 but only 2 iterations

4.2 Related Rates

Exercises 7, 9, 11, 15

4.3 Optimization

Exercises 8, 9, 11, 12, 13, 15, 18

4.4 Differentials

Exercises 7-14, 31-34

5.1 Antiderivatives and Indefinite Integration

Exercises 9-27, 31-39

5.2 The Definite Integral

Exercises 5-26

5.3 Riemann Sums

Exercises 29-34