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Calculus I Limits and Derivatives

Sunil Karna

Section 1.1 Review

Evaluate:
  1. \(\displaystyle (-3)^4\)
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  2. \(\displaystyle 3^{-4}\)
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  3. \(\displaystyle \left(\frac{3x^{3/2}y^3}{x^2y^{-1/2}}\right)^{-2}\)
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Simplify
  1. \(\displaystyle 3(x-5) +5(2x+3)\)
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  2. \(\displaystyle (x+3)^3\)
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  3. \(\displaystyle (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})\)
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Factor each expression
  1. \(\displaystyle 4x^2-25\)
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  2. \(\displaystyle 2x^2+5x-12\)
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  3. \(\displaystyle x^3-3x^2-4x+12\)
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  4. \(\displaystyle x^4+27x\)
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  5. \(\displaystyle \frac{x^2+3x+2}{x^2-x-2}\)
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  6. \(\displaystyle \frac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{y}-\frac{1}{x}}\)
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Rationalize the expression
  1. \(\displaystyle \frac{\sqrt{10}}{\sqrt{5}-2}\)
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  2. \(\displaystyle \frac{\sqrt{4+h}-2}{h}\)
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Solve the following equations
  1. \(\displaystyle x+5=14-\frac{1}{2}x\)
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  2. \(\displaystyle \frac{2x}{x+1}=\frac{2x-1}{x}\)
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  3. \(\displaystyle x^2-x-12\)
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  4. \(\displaystyle 3|x-4|=10\)
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Subsection 1.1.1 Quadratic Equations

\begin{equation*} ax^2+bx+c=0; \, \quad a\neq 0 \end{equation*}
Quadratic equation always has two roots, \(\alpha\) and \(\beta.\) The roots of quadratic equation can be found by using formula:
\begin{equation*} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{equation*}
Here \(b^2-4ac =D\) is called Discriminant and the roots
\begin{equation*} \alpha =\frac{-b + \sqrt{b^2-4ac}}{2a} \end{equation*}
and
\begin{equation*} \beta =\frac{-b - \sqrt{b^2-4ac}}{2a} \end{equation*}
\(\alpha + \beta = \frac{-b}{a}\) and \(\alpha \beta = \frac{c}{a}\text{.}\) If Discriminant, \(D \gt 0\) then the two roots are real and unequal and the graph of quadratic function crosses the x-axis at two points. If Discriminant, \(D = 0\) then the two roots are real and equal and the graph of quadratic function touches the x-axis. If Discriminant, \(D \lt 0 \) then the two roots are complex and the graph of quadratic function does not touches the x-axis. If the coefficient of \(x^2\text{,}\) \(\, a \gt 0 \) then graph is upward open parabola, if \(a \lt 0 \) then it is downward looking parabola. Draw graph of the following quadratic function and solve the equation using factorization method if possible.
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Desmos
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www.desmos.com/calculator
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  1. \begin{equation*} x^2-6x+9 =0 \end{equation*}
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  2. \begin{equation*} x^2+10x+25 =0 \end{equation*}
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  3. \begin{equation*} 4x^2-20x+25 =0 \end{equation*}
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  4. \begin{equation*} x^2-2x+5 =0 \end{equation*}
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  5. \begin{equation*} 2x^2+3x-2 =0 \end{equation*}
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  6. \begin{equation*} 2x^2+5x+3 =0 \end{equation*}
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  7. \begin{equation*} 3x^2-11x+6 =0 \end{equation*}
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  8. \begin{equation*} x^2+9=0 \end{equation*}
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  9. \begin{equation*} x^2+10x+13 =0 \end{equation*}
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  10. \begin{equation*} x^2+3x-10 =0 \end{equation*}
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  11. \begin{equation*} x^2-5x+4 =0 \end{equation*}
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  12. \begin{equation*} x^2-x-6 =0 \end{equation*}
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Subsection 1.1.2 Number Systems

A number is a mathematical value used for counting, measuring, or labelling objects. In our daily life, we use these numbers to perform all the mathematical operations like addition, subtraction, multiplication, and division.
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  1. Natural Numbers:
    \begin{equation*} 1,2,3,4,5,6,7,8,9,10,11,\ldots, \infty \end{equation*}
    Natural numbers starts from 1 which is the first number to begin counting and continue endlessly. It is also called counting numbers. Each number in natural number can be obtained by adding 1 to the previous number. It may contain 10, 20, 100, etc. but not mare zero. Natural numbers are all positive numbers.
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  2. Whole Numbers:
    \begin{equation*} 0, 1,2,3,4,5,6,7,8,9,10,11,\ldots, \infty\text{.} \end{equation*}
    Whole numbers also contain zero, (\(0\)) and every numbers in natural numbers. Whole numbers are also all positive.
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  3. Integers:
    \begin{equation*} -\infty, \ldots -5,-4,-3,-2, -1, 0, +1,+2,+3,+4,+5,\ldots, \infty\text{.} \end{equation*}
    Integers are the Whole numbers with both positive and negative numbers. Hence, we have positive inegers as \(0, +1,+2,+3,+4,+5,\ldots \infty \) and negative integers as \(-1,-2,-3, \ldots, -\infty \text{.}\)
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  4. Rational Numbers: Rational numbers can be expressed as a fraction in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are any integer but \(q \neq 0\text{.}\) Rational numbers are positive, negative, and zero.
    \begin{equation*} \frac{p}{q} = \frac{5}{2}, -\frac{5}{1}, \frac{1}{2} =0.5, -\frac{1}{3} =-0.33333, \frac{1}{3}, \frac{7}{8}, \frac{8}{4}, \sqrt{9}, 0, \ldots. \end{equation*}
    Rational numbers can be converted to both terminating and non-terminating decimal numbers but with repeating value. Examples:
    \begin{equation*} \frac{25}{4} = 6.25. \text{ (terminating value)} \end{equation*}
    \begin{equation*} \frac{1}{3} = 0.333 \ldots = 0.\dot{3} = 0.\bar{3}. \text{ (3 repeats forever)} \end{equation*}
    \begin{equation*} \frac{1}{7} = 0.142857142857 \ldots. \text{ (14285714 repeats forever)} \end{equation*}
    \begin{equation*} \frac{77}{600} = 0.128333 \ldots. \text{ (3 repeats forever)} \end{equation*}
    To show a repeating digits in a decimal number, we put either a dot or a line above the repeating digit as shown above.
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    Note: The sum of two rational numbers or the product of two rational numbers is also rational. Examples:
    \begin{equation*} \frac{1}{2} + \frac{1}{3} = \frac{5}{6} (\text{rational}) \end{equation*}
    \begin{equation*} \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} (\text{rational}) \end{equation*}
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  5. Irrational Numbers: The numbers which are not rational numbers are called irrational numbers. Irrational numbers could be written in decimals but not in the form of fractions, which means they cannot be written as the ratio of two integers. Irrational numbers are non-terminating and non-repeating digits after decimal point. Examples:
    \begin{equation*} \sqrt{3} = 1.73205 \ldots. \end{equation*}
    \begin{equation*} \sqrt{8} = 2.8284271 \ldots. \end{equation*}
    \begin{equation*} \pi = 3.1415926 \ldots. \end{equation*}
    \(\sqrt{2}, \sqrt{10}, 1.24387\ldots, \ldots \) are irrational numbers.
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    Note: The sum of two irrational numbers or the product of two irrational numbers is not always irrational. Examples:
    \begin{equation*} \sqrt{2}+ \sqrt{2} = 2\sqrt{2} (\text{irrational}) \end{equation*}
    \begin{equation*} \sqrt{5} + (-\sqrt{5}) = 0 (\text{rational}) \end{equation*}
    \begin{equation*} \sqrt{2}\times \sqrt{3} = \sqrt{6} (\text{irrational}) \end{equation*}
    \begin{equation*} \sqrt{2}\times \sqrt{2} = \sqrt{4} (\text{rational}) \end{equation*}
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    Rational numbers are finite and repeating decimals whereas irrational numbers are infinite and non-repeating.
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    Some irrational numbers are also special numbers like Pi (\(\pi\)), Euler number (\(e\)), golden ratio (\(\phi\)), and square root of 2 (\(\sqrt{2}\)).
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  6. Real Numbers: A real number is any rational or irrational number, including positive and negative whole numbers, integers, decimals, fractions, and numbers such as pi \((\pi)\) and Euler’s number \((e)\text{.}\)
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    Infinity \((\infty)\) and negative infinity \((-\infty)\) are not real numbers. Because infinity and negative infinity can have different values. For example, the set of whole numbers is infinite. So is the set of integers. But, the two sets are not of the same size.
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    The four main properties of real numbers are the commutative property, associative property, distributive property, and identity property. If \(m, n,\) and \(r\) are real numbers, then:
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  7. Complex Numbers: The complex number is the combination of a real number and an imaginary number. The complex number is in the form of \(a+ib\text{,}\) where \(a\) = real number and \(ib\) = imaginary number. Also, \(a,b\) belongs to real numbers and \(i = \sqrt{-1}.\) This numbers are very useful in analysing phasor diagram in electricity.
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    The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as \(Im()\text{.}\) Example: \(\sqrt{-2}, \sqrt{-7}, \sqrt{-11}\) are all imaginary numbers. The roots of equation \(x^2+1 = 0\) is an imaginary number.
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Subsubsection 1.1.2.1 Standard Notations of Number Systems:

\begin{align*} {\mathbb N} &= \{n: n \text{ is a natural number}\} = \{1, 2, 3, 4, 5, \ldots \}\\ {\mathbb W} &= \{w: w \text{ is a whole number}\} = \{0, 1, 2, 3, 4, 5, \ldots \}\\ {\mathbb Z} &= \{n : n \text{ is an integer} \} = \{\ldots, -2, -1, 0, 1, 2, \ldots \}\\ {\mathbb Q} &= \{r : r \text{ is a rational number}\} = \{p/q : p, q \in {\mathbb Z} \text{ where } q \neq 0\}\\ {\mathbb R} &= \{ x : x \text{ is a real number} \} = \{x :x \text { is rational or irrational number.} \}\\ {\mathbb E} &= \{ x : x \text{ is an even number} \} = \{\ldots, -6, -4, -2, 0, 2, 4, 6, \ldots \} \\ {\mathbb O} &= \{ x : x \text{ is an odd number} \} = \{\ldots, -3, -1, 1, 3, 5, \ldots \} \\ {\mathbb C} &= \{z : z \text{ is a complex number}\}\text{.} \end{align*}
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The symbol ’\(\in\)’ read as ’belongs to’ or ’is an element of’, where as ’\(\notin\)’ stands for ’does not belong to’ or ’is not an element of’. Example:
\begin{equation*} A = \{ 2,4,8,10,12\}; 8 \in A; \text{ but } 14 \notin A. \end{equation*}
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