Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
![Here we apply the highest-power rule for polynomials
introduced in Example 1: the curve behaves like y
when r 0 and when r →-0o. The graph is sketched in Figure 8.3.
a for very large |rl, so y ∞
3
-22
Figure 8.3: y = x - 4r3 +5
Exercises
8.1.1 In Exercise 4.3.6, you were asked to sketch in one diagram the graphs of y = r²,
y = r* and y = x* for all r in R. We can now use the techniques of this section
to confirm the shape of each graph.
Show that the graph of y = x² has a minimum point at (0,0) and that the graph
of y = r³ has a critical point of inflexion at (0,0). What about the graphs of
y = x, y = x³5 and so on?
8.1.2 Find the critical points of
y = x° – 12x2 + 21x +1
and determine their nature. Sketch the curve.
8.1.3 Find the critical values of y = – 2x and determine their nature. Sketch the
curve.
8.2 The second derivative
If f is a differentiable function, then f' is a function, which may itself be differentiable.
If it is, we denote the derivative of f'(x) with respect to x by f"(x) (read "eff-double-
prime r") and call it the second derivative of f (x). The function f is then said to be
twice differentiable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9378ef80-7767-4c68-bae6-da61c50171e5%2Fe0967a2e-5681-4960-b738-39aa6a3e33c8%2Foxhua79_processed.jpeg&w=3840&q=75)
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