Tangents and Normals of Curves
If you differentiate the equation of a curve, you will get a formula
for the gradient of the curve. Before you learnt calculus, you would
have found the gradient of a curve by drawing a tangent to the curve
and measuring the gradient of this. This is because the gradient of a
curve at a point is equal to the gradient of the tangent at that
point.
Example:
Find the equation of the tangent to the curve y = x³ at the point (2,
8).
dy = 3x²
dx
Gradient of tangent when x = 2 is 3×2² = 12.
From the coordinate geometry section, the equation of the tangent is
therefore:
y - 8 = 12(x - 2)
so y = 12x - 16
You may also be asked to find the gradient of the normal to the curve.
The normal to the curve is the line perpendicular (at right angles) to
the tangent to the curve at that point.
Remember, if two lines are perpendicular, the product of their
gradients is -1.
So if the gradient of the tangent at the point (2, 8) of the curve y =
x³ is 8, the gradient of the normal is -1/8, since -1/8 × 8 = -1.
Integration
Introduction
Integration is the reverse of differentiation.
If y = 2x + 3, dy/dx = 2
If y = 2x + 5, dy/dx = 2
If y = 2x, dy/dx = 2
So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc.
For this reason, when we integrate, we have to add a constant. So the
integral of 2 is 2x + c, where c is a constant.
A 'S' shaped symbol is used to mean the integral of, and dx is written
at the end of the terms to be integrated, meaning 'with respect to x'.
This is the same 'dx' that appears in dy/dx .
To integrate a term, increase its power by 1 and divide by this
I have plotted graphs from both sets of calculated gradients however I will concentrate on the graph plotted from the results show above as
So using this formula but with the data we collected from our first attempt, this is what it would look like; Tan(60°) x 23m = 39m. As you can tell this answer collected from our first attempt is very well incorrect, but at the time, our group did not know this.
The spreadsheet from the eLearn titled “Experiment I data” was downloaded to the computer. There were three inhibitors data given. The slope (V0) was calculated for each inhibitor data using the time versus response. The formula used to calculate slope was typed = slope (B6:B11, $A$6: $A$11) in the cell right below the last Reponses, and then from that cell dragged horizontally to get the rest of the slope. This step was repeated for Inhibitor 2, and 3 data.
If I were using a cut out of length 1cm, the equation for this would
the root to the function, like if it is a parabola with its vertex is placed
The Relationship between the Angle of Elevation of a Ramp and the Speed of a Ball
slope. I think that out of all the variables, this is the one which is
In Newtonian physics, free fall is any motion of a body where its weight is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it and it moves along a geodesic. The present article only concerns itself with free fall in the Newtonian domain.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
Integration and differentiation are systems that stand for general dialectical processes of constant and change. An integrated system is which an individual part is that cover the system true form successfully interconnected and reinforced. To integrate an item means to organize and incorporate different parts creatively breaking the rules to somehow make something that originally separate work great together. The differentiate system has a unique function that cannot be changed or molded. To differentiate is to be bias toward different parts that are unique to themselves.
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
and 8 can be written as 2 , while 5, 6, and 7 can be written using some
whereβ the intercept 0 and β the slope 1 are unknown constants and ε is a random error component .
... resultant speed and, by the definition of the tangent, to determine the angle of which the object is launched into the air.