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Pre calculus

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Precalculus is a widespread concept as it covers large number of topics and other theories. Precalculus does not include topics from calculus. But, it covers the topics and concepts which will be useful in calculus. Precalculus is studied from primary schools to research schools to help students in understanding important topics that will be used to solve problems in calculus. So, we can say it teaches the basics of calculus. Calculus is the branch of mathematics used to study any phenomena involving change. Precalculus is also known as introduction to analysis. Precalculus deals with trigonometry and explains the concepts which are introductory to calculus.
Precalculus is a preparation for calculus.

Precalculus Topics

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Some important concepts of precalculus are as follows:
  • Curve: Plotting equations on a graph and the shape formed is called the curve.
  • Polar coordinates: This is the system of coordinates in which coordinates are defined with the help of angles.
  • Plane: Plane can be defined as a two dimensional surface which defines linear equations.
  • Tangents: Tangent is a line which touches the outer surface of the circle or circular object at exactly one point.
  • Complex Numbers: Complex numbers are the numbers which consist of two parts: one is the real part and the other is the imaginary part.
  • Conic Sections: Conic sections are the curves which are generated, when a plane intersects or cuts the parts of a cone. There are different types of conic sections like ellipse, hyperbola, and parabola.
  • Logarithms: Logarithm of a number is the exponent to which another fixed value, the base must be raised to produce that number.
  • Natural Logs: These are the logarithms which have base ‘e’.
  • Functions: A function consists of variables with operations defined on them and these functions depend on particular variables.
  • Vectors: Vectors can be defined as quantity which has both magnitude and direction.
  • Limit: Limit is a value that a function approaches as the input approaches some value. Usually abbreviated as lim.
  • Real Numbers: Real number is a value that represents a quantity along a continuous line.
  • Trigonometry: Branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides.
  • Complex Analysis: A branch of mathematical analysis that investigates functions of complex numbers.
  • Mathematical Induction: A method of mathematical proof used to establish that a given statement is true for all natural numbers.

Precalculus Problems

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Given below are some of the problems in Precalculus.

Solved Examples

Question 1: Simplify $3^{-4}$
Solution:
$3^{-4}$ = $\frac{1}{3^{4}}$
 = $\frac{1}{81}$
Therefore, $3^{-4}$ = $\frac{1}{81}$

Question 2: Multiply (15 + i10) and (12 + i7)
Solution:
(15 + i10) x (12 + i7) = 15(12 + i7) + i10(12 + i)

= 180 + i105 + i120 + i$^2$10

= 180 + i225 - 10

= 170 + i225

=> (15 + i10) x (12 + i7) = 170 + i225

Question 3: Solve : $\lim_{x\rightarrow 2}$ $\frac{5x^{2}+2x -5}{3-2x}$
Solution:
$\lim_{x\rightarrow 2}$ $\frac{5x^{2}+2x -5}{3-2x}$ = $\frac{\lim_{x\rightarrow 2}(5x^{2}+2x-5)}{\lim_{x\rightarrow 2}(3-2x)}$

= $\frac{\lim_{x\rightarrow 2}5x^{2}+\lim_{x\rightarrow 2}2x - \lim_{x\rightarrow 2} 5}{\lim_{x\rightarrow 2}3 -\lim_{x\rightarrow 2}2x }$

= $\frac{5(2)^{2}+ 2(2)-5}{3 - 2(2)}$

= $\frac{20+4-5}{3-4}$

=  - 19