Calculus Made Easy

This course builds on students' previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.

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1.3 - Properties of Limits

18 November, 2021 - yolo_keurig | 1 Comments

Now that we have understood what the limit of a function is, we must now know the various properties surrounding the limits of a function.

There are 7 main properties of limits that we must know in calculus and vectors.

lim k = k, for all constants of k
x->a

lim x = a
x->a

lim [f(x) +/- g(x)] = lim f(x) +/- lim g(x)
x->a x->a x->a

lim [cf(x)] = c[lim f(x)], for all constants of c
x->a x->a

lim [f(x)g(x)] = [lim f(x)] [lim g(x)]
x->a x->a x->a

lim f(x) = lim f(x), given that lim g(x) does not equal to 0
x->a g(x) x->a x->a
lim g(x)
x->a
(The property above this statement is meant to represent what to do when two limits are being divided with each other.)

lim [f(x)]^n = [lim f(x)]^n, for any value of n that is a rational number.
x->a x->a

Due to constraints regarding Steam's implementation of textboxes (almost seems like they weren't made for mathematical calculations,) our example will unfortunately be very simple. However, I urge those of you intrigued enough to read to here to attempt problems involving these properties yourself.

Evaluate lim (3x^2+ 4x - 1)
x->2
Solution :
= lim (3x^2) + lim (4x) - lim (1)
x->2 x->2 x->2

= 3 lim (x^2) + 4 lim (x) - 1
x->2 x->2

= 3lim x ^2 +4(2) - 1
x->2

= 3(2)^2+8-1
= 19

When we are solving for limits algebraically, we can use the techniques of:
- Direct Substitution
- Factoring
- Rationalizing
- One-sided limits
- Change of variables

For these techniques, you may also use a graph and/or a table in order to check your result.

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18 November, 2021 - yolo_keurig | 1 Comments

Thank you for reading our stupid reviews I hope in the future Calculus and Vectors will grow to finally become a military force and allow me to finally achieve total world domination.

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