calculus problems and answers

The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieve..

Let f(x) be a continuous function on the closed interval [a, b]. Let the area function A(x) be defined by th..

Let f(x) be a continuous function defined on an interval [a,b]. between the limits a and b. This statement is also known as 'fundamental theorem of calculus'. We call b, the upper limit of x and a, the lower limit. If in place of F(x) we take F(x)+c as the value of the integral, we have =..

y 2 = 4ax (1) Differentiating with respect to x Substitute for 4a in (1), we get y - 2xy' = 0 Note that the given equation is differentiated only once to obtain the differential equation since it has only one consta..

Because of these condition, the 2 n d order differential equation y''= 2 has particular solution x 2 + x + 2. The values f (0) = 2 and f '(0) = 1 are called initial values. The problem of finding the solution of a differential equation that satisfies these prescribed initial conditions is..

Let y = vx v = log x +c Resubstituting for v, we get..

Let y = log u, u = tan t, t = x/2 From the above example, it is clear that chain rule can be extended. Let v = f(u), u = g(x), x = h(t) Then, we have..

The function where D x is a small change in x. = 2x - 9 f '(100) = 2 (100) - 9 = ..

Let Differentiating with respect to x, we get ..

Differentiating w.r.t t, we get Differentiating both sides with respect to x, we get ..