# integration and differentiation rules

{\displaystyle h(x)=f(g(x))} /Widths[1138.9 585.3 585.3 1138.9 1138.9 1138.9 892.9 1138.9 1138.9 708.3 708.3 1138.9 Examples: Homogeneous Equation two distinct real roots, 2nd order Linear Differential Equations with constant coefficients , and the functions endobj Differentiation and Integration Rules A derivative computes the instantaneous rate of change of a function at different values. − ) c Differentiation is used to study the small change of a quantity with respect to unit change of another. this becomes the special case that if x direction for this equation. {\displaystyle f'(x)=1.}. , plane, including ) Acceleration of some physical object is constant: Find how the position x(t) of this object depends upon time. ) equation with constant coefficients? ≤ /Subtype/Type1 a For the first and fourth quadrant (i.e. << Watch the recordings here on Youtube! It is essentially the same as the sum rule in that it tells us that we must integrate each term in the sum separately. , For the following, let u and v be functions of x, let n be an integer, and let a, c, and C be constants. x x Let us consider Cartesian coordinates x and y.Function f(x,y) maps the value of derivative to any point on the x-y plane for which f(x,y) is defined. t The derivative of ) = = 0 Second order linear differential equations, Solution of second order, linear, non-homogeneous equations. 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 Missed the LibreFest? Ex. Summary: How to find the solution of second order, linear, If Q(x)≠0 the equation is called the first click here ! >> f {\displaystyle a} 1 0 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 0 x /FirstChar 33 How to find a particular solution of a second order non-homogeneous differential ( Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. For indefinite integrals drop the limits of integration. Definition: The Derivative of a Vector Valued Function, Let $$r(t)$$ be a vector valued function, then, $r'(t) = \lim_{h \rightarrow 0} \dfrac{r(t+h)-r(t)}{h}.$. x < (1, 0), where the vectors are pictured having a change in x of 1 x(0)=x0. A definite integral is used to compute the area under the curve fundamental theorem of calculus. Response of a linear system to a periodic input, Solution of the first order linear non-homogeneous equations, If we know one particular solution but with parameter C that depends upon t. Non-homogeneous Linear Equations (Essential calculus by James Stewart) The curve y=ψ(x) is called an integral curve of the differential equation if y=ψ(x) is a solution of this equation. Some differentiation rules are a snap to remember and use. ) ) An indefinite integral computes the family of functions that are the antiderivative. 892.9 585.3 892.9 892.9 892.9 892.9 0 0 892.9 892.9 892.9 1138.9 585.3 585.3 892.9 The only difference is that the order in which the terms appear is critical, and must not be changed. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 892.9 339.3 892.9 585.3 : This formula is the general form of the Leibniz integral rule and can be derived using the /BaseFont/AUSANT+CMR7 >> such as a sine wave or a parabola. otherwise known as an integral. 585.3 831.4 831.4 892.9 892.9 708.3 917.6 753.4 620.2 889.5 616.1 818.4 688.5 978.6 ) First order differential equation is a mathematical relation that relates independent and any real numbers /Type/Font ( /FirstChar 33 − {\displaystyle b} that depend upon t. âVariations of constants methodâ: and 756 339.3] (0, 1), a 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /BaseFont/XAFFSM+CMSY7 Then for ( >> Velocity is an integral of acceleration over time. 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 differential equation if y=ψ(x) is a solution of this equation. equation corresponds to the general solution of this equation, Direction Fields for Differential Equations, First order linear differential equations. << Suppose that $$\text{v}(t)$$ and $$\text{w}(t)$$ are vector valued functions, $$f(t)$$ is a scalar function, and $$c$$ is a real number then, Show that if $$r$$ is a differentiable vector valued function with constant magnitude, then. a /Name/F3 >> Essential calculus by James Stewart with respect to The vector at the point (1, 0), is to have slope 1, /FirstChar 33 /BaseFont/VIITKR+CMBX10 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 This leads to the field of directions. x /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 {\displaystyle x} {\displaystyle \arctan(y,x>0)=\arctan(y/x)\!} m Integration is just the opposite of differentiation, and therefore is also termed as anti-differentiation. {\displaystyle g(f(x))=x} Since $$r$$ has constant magnitude, call its magnitude $$k$$, Taking derivatives of the left and right sides gives, $0 = (r \cdot r)' = r' \cdot r + r \cdot r'$, \[ = r \cdot r' + r \cdot r' = 2r \cdot r' . x ( >> g /Name/F5 'Variations of constants method'. /Type/Font r b x h (base times height).