![]() Find d h / dt or the velocity during upward motion and descent of the ball. 2x = 4x Example 5 Find dy/dx of y=2 + 3x + 4 dy/dx = 4x + 3 -Applied Problem 1: Brian throws a ball up with some force.The height reached by a ball with initial velocity of 64 feet per second and intitial height of 6 feet from Brian's hand is given by the equation: h (t) = - 16 + 64 t + 6 where t is time in seconds. Let us now proceed to find dy/dx for other functions: 3 y = f(x) = Let us go through the steps as for the previous function: - Subtract-> Divide by h -> f ( x+h) - f (x) = 2xh + f(x+h) - f (x) - = 2x + h h Limit ( f(x+h) - f(x) ) /h = 2x h -> 0 Therefore: dy/dx = 2x when y = f(x) = Some problems: Example 4 Find dy/dx or the derivative of Y = 2 dy/dx= (2). Then dy/dx = k dy1/dx Let me illustrate these results with examples: Example1 Find dy/dx or derivative of y = 4x dy/dx = 4 Example 2 Find dy/dx when y = 4x + 6x dy/dx = 4 + 6= 10 Example 3 Find the derivative dy/dx for the straight line : y = 3x + 4 dy/dx = 3 + 0 = 3 since 4 is a constant. Suppose we have a function : y = k f(x) = ky1 where k is a constant. Then, differential coefficent is just this: d (y)/d(x) = (y1 - y) / to zero. y = f (x) y1= f ( x + x) where x is a small change in x. In other words, what is the change in y when x is changed by an tiny amount. Differential Calculus and Integral Calculus are like two opposing processes.In differential calculus, for a function Y = f(x), we take smaller and smaller intervals of x and take into account the corresponding changes in y. But remember to add C.C alculus Tutorial-1 Differential C alculus Introduction Calculus is divided into two parts: Differential Calculus and Integral Calculus.There is also a related subject- Differential Equations. If we are lucky enough to find the function on the result side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. Which teaches us to always remember "+C". ![]() And the increase in volume can give us back the flow rate.The flow still increases the volume by the same amount.The derivative of the volume x 2+C gives us back the flow rate:Īnd hey, we even get a nice explanation of that "C" value. The integral of the flow rate 2x tells us the volume of water: Derivative: If the tank volume increases by x 2, then the flow rate must be 2x.Integration: With a flow rate of 2x, the tank volume increases by x 2.Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap):Īs the flow rate increases, the tank fills up faster and faster: This shows that integrals and derivatives are opposites! We can integrate that flow (add up all the little bits of water) to give us the volume of water in the tank. The input (before integration) is the flow rate from the tap. So we wrap up the idea by just writing + C at the end. So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value. ![]() and the derivative of x 2+99 is also 2x,īecause the derivative of a constant is zero.and the derivative of x 2+4 is also 2x,.It is there because of all the functions whose derivative is 2x: The symbol for "Integral" is a stylish "S"Īfter the Integral Symbol we put the function we want to find the integral of (called the Integrand),Īnd then finish with dx to mean the slices go in the x direction (and approach zero in width). Integration can sometimes be that easy! Notation That simple example can be confirmed by calculating the area:Īrea of triangle = 1 2(base)(height) = 1 2(x)(2x) = x 2
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