A Mathematical Definition. The difference quotient is the slope of the secant line between two points. The difference quotient can be used to find the slope of a curve, as well as the slope of a straight line. After we find the difference quotient of a function, we have a new function, called the derivative.
Correspondingly, is the difference quotient the same thing as the derivative?
The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)).
What is the normal of a line?
The derivative of a function has many applications to problems in calculus. The derivative of a function at a point is the slope of the tangent line at this point. The normal line is defined as the line that is perpendicular to the tangent line at the point of tangency.
What is the secant line?
In geometry, a secant line commonly refers to a line that intersects a circle at exactly two points (Rhoad et al. 1984, p. 429). There are a number of interesting theorems related to secant lines.
What does the difference quotient do?
For a function f, the formula . This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative.
How do you know if a function is even or odd?
You may be asked to “determine algebraically” whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f (–x) = f (x), so all of the signs are the same), then the function is even.
What makes a function odd or even?
DEFINITION. A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f. A function f is odd if the graph of f is symmetric with respect to the origin.
Is a sine function even or odd?
Sine is an odd function, and cosine is an even function. You may not have come across these adjectives “odd” and “even” when applied to functions, but it’s important to know them. A function f is said to be an odd function if for any number x, f(–x) = –f(x).
Is a Cosecant function even or odd?
From the graphs above, we see that tan, cot and csc are odd functions while sec is an even function. These symmetries give us the following identities. These identities may be obtained directly from the symmetries of the sine and cosine functions.
Is a sine function symmetric?
The cosine is known as an even function, and the sine is known as an odd function. Generally speaking, Some functions are odd, some are even, and some are neither odd nor even. If a function is even, then the graph of the function will be symmetric with the y-axis.
Is Tan symmetric?
Sketch the graphs of secant and cosecant functions. The tangent function is odd, tan(–x) = –tan x. The graph of y = tan x is symmetric with respect to the origin. tan x = sin x/cos x tangent is undefined for values at which cos x = 0.
Is cot an even or odd function?
Students should know that cosine and secant are even functions and are symmetric with respect to the y-axis. We know this is true because of the negative angle identities for cosine and secant. As expected, the rest of ’em (sine, cosecant, tangent, and cotangent) are odd functions and are symmetric to the origin.
Is cot odd or even?
Are Secant, Tangent, Cosecant, and Cotangent Even or Odd? We can determine whether each of the other basic trigonometric functions is even, odd, or neither, with just these two facts and the reciprocal identities. Thus tangent takes the form f ( − x ) = − f ( x ) , so tangent is an odd function.
Is a tangent function even or odd?
Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. Sine and tangent are both odd functions, and cosine is an even function. In other words, sin(–x) = –sin x.
What is a function that is even?
A function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x). See also. Odd function.
What are the six basic trigonometric functions?
The 6 Trig Ratios. For any right triangle, there are six trig ratios: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Here are the formulas for these six trig ratios: Given a triangle, you should be able to identify all 6 ratios for all the angles (except the right angle).
What are the six trigonometric functions?
There are six main trigonometric functions:
What is sine cosine and tangent?
Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Before getting stuck into the functions, it helps to give a name to each side of a right triangle: “Opposite” is opposite to the angle θ “Adjacent” is adjacent (next to) to the angle θ
What do the letters in Sohcahtoa stand for?
Sine (Opposite over Hypotenuse), Cosine (Adjacent over Hypotenuse), Tangent (Opposite over Adjacent) SOHCAHTOA. Some Old Horse Caught Another Horse Taking Oats Away (mnemonic for remembering sine, cosine and tangent)
What is cos over sin?
Each of the six trig functions is equal to its co-function evaluated at the complementary angle. Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2π while tangent and cotangent have period π. Identities for negative angles.
What is equal to tan?
A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent). The tangent is implemented in the Wolfram Language as Tan[z].
What is sin tan?
Sin, Cos and Tan. This section looks at Sin, Cos and Tan within the field of trigonometry. A right-angled triangle is a triangle in which one of the angles is a right-angle. The adjacent side is the side which is between the angle in question and the right angle.
How do you find sin?
Step 2 Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question. Step 3 For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse or for Tangent calculate Opposite/Adjacent. Step 4 Find the angle from your calculator, using one of sin-1, cos-1 or tan.
What is CSC?
A Common Service Center (CSC) is an information and communication technology (ICT) access point created under the National e-Governance Project of the Indian government. The project plan includes the creation of a network of over 100,000 CSCs throughout the country.