How To Understand Calculus With Pictures

Functions are often written as f(x)=x+3. {\displaystyle f(x)=x+3. } This means that the function f(x){\displaystyle f(x)} always adds 3 to the number you input for x. {\displaystyle x. } If you want to input 2, write f(2)=(2)+3,{\displaystyle f(2)=(2)+3,} or f(2)=5. {\displaystyle f(2)=5. } Functions can map complex motions too. NASA, for example, has a function that describes how fast a rocket will go based on how much fuel it burns, the wind resistance, and the weight of the rocket itself.

Limits are easiest to see on a graph – are the points that a graph almost touches, for example, but never does? Limits can be a number, infinity, or not even exist. For example, if you add 1 + 2 + 2 + 2 + 2 + . . . forever, your final number would be infinitely large. The limit would be infinity.

Algebra. Understand different processes and be able to solve equations and systems of equations for multiple variables. Understand the basic concepts of sets. Know how to graph equations. Geometry. Geometry is the study of shapes. Understand the basic concepts of triangles, squares, and circles and how to calculate things like area and perimeter. Understand angles, lines, and coordinate systems Trigonometry. Trigonometry is branch of maths which deals with properties of circles and right triangles. Know how to use trigonometric identities, graphs, functions, and inverse trigonometric functions.

Many smartphones and tablets now offer cheap but effective graphing apps if you do not want to buy a full calculator.

Finding instantaneous change is called differentiation. Differential calculus is the first of two major branches of calculus.

Acceleration is a derivative – it tells you how fast something is speeding up or slowing down, or how the speed is changing.

The slope of a line is the change in y divided by the change in x. The bigger the slope, the steeper a line. Steep lines can be said to change very quickly. Review how to find the slope of a line if your memory is hazy.

For example, in y=x2,{\displaystyle y=x^{2},} you can take any two points and get the slope. Take (1,1){\displaystyle (1,1)} and (2,4). {\displaystyle (2,4). } The slope between them would equal 4−12−1=31=3. {\displaystyle {\frac {4-1}{2-1}}={\frac {3}{1}}=3. } This means that the rate of change between x=1{\displaystyle x=1} and x=2{\displaystyle x=2} is 3.

For example, scientists study how quickly some species are going extinct to try to save them. However, more animals often die in the winter than the summer, so studying the rate of change across the entire year is not as useful – they would find the rate of change between closer points, like from July 1st to August 1st.

Think of the example where you keep dividing 1 by 2 over and over again, getting 1/2, 1/4, 1/8, etc. Eventually you get so close to zero, the answer is “practically zero. " Here, your points get so close together, they are “practically instantaneous. " This is the nature of derivatives.

There are different notations for derivatives. In the previous step, derivatives were labeled with a prime symbol – for the derivative of y,{\displaystyle y,} you would write y′. {\displaystyle y^{\prime }. } This is called Lagrange’s notation. There is also another popular way of writing derivatives. Instead of using a prime symbol, you write ddx. {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}. } Remember that the function y=x2{\displaystyle y=x^{2}} depends on the variable x. {\displaystyle x. } Then, we write the derivative as dydx{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}} – the derivative of y{\displaystyle y} with respect to x. {\displaystyle x. } This is called Leibniz’s notation.

How fast does the marble change location? What is the rate of change, or derivative, of the marble’s movement? This derivative is what we call “speed. ” Roll the marble down an incline and see how fast in gains speed. What is the rate of change, or derivative, of the marble’s speed? This derivative is what we call “acceleration. ” Roll the marble along an up and down track like a roller coaster. How fast is the marble gaining speed down the hills, and how fast is it losing speed going up hills? How fast is the marble moving exactly halfway up the first hill? This would be the instantaneous rate of change, or derivative, of that marble at its one specific point.

Making geographic models and studying volume is using integration. Integration is the second major branch of calculus.

Imagine you are adding together a lot of little slices under the graph, and the width of each slice is ‘’almost’’ zero.

The first symbol, ∫,{\displaystyle \int ,} is the symbol for integration (it is actually an elongated S). The second part, f(x),{\displaystyle f(x),} is your function. When it is inside the integral, it is called the integrand. Finally, the dx{\displaystyle \mathrm {d} x} at the end tells you what variable you are integrating with respect to. Because the function f(x){\displaystyle f(x)} depends on x,{\displaystyle x,} that is what you should integrate with respect to. Remember, the variable you are integrating is not always going to be x,{\displaystyle x,} so be careful what you write down.

Integrate by substitution. Calculate indefinite integrals. Integrate by parts.

For example, remember that the derivative of speed is acceleration, so you can use speed to find acceleration. But if you only know the acceleration of something (like objects falling due to gravity), you can integrate it to find the speed!

This lets you find the volume of any solid in the world, as long as you have a function that mirrors it. For example, you can make a function that traces the bottom of a lake, and then use that to find the volume of the lake, or how much water it holds.