CALCULUS Areas of Focus: 1. Differentiation 2. Maximization and minimization 3. Partial derivatives 4. Integration 5. Integration over a line 6. Double integrals 7. Integration over an area 8. Centroid of an area 9. Integrating differential equations Differentiation: The derivative of a function is a measure of how the function changes as a result of a change in the value of its argument. Given the function f(x), the derivative of f with respect to x is written as or as , and is defined by As shown in the figure, the derivative of the function f(x) at point x gives the slope of the function at x in terms of the ratio of the rise divided by the run for the line AB that is tangent to the curve at point x. Page 1 of 23 The derivative of the function f(x) is also sometimes written as f ' (x). As shown in the figure, one can also write the definition of the derivative as The basic rules of differentiation are The derivative of commonly used functions: The following is a list of the derivatives of some of the more commonly used functions. Page 2 of 23 The product rule for derivatives: Consider a function such as f(x)=g(x)h(x) that is the product of two functions. The product rule can be used to calculate the derivative of f with respect to x. The product rule states that For example, To take the derivative of f(x)=(2x+3)(4x+5)2 one can follow these steps The chain rule for derivatives: Consider the function f(U), where U is a function of x. One can calculate the derivative of f with respect to x by using the chain rule given by Page 3 of 23 For example, to calculate the derivative of f(U) = Un with respect to x, where U = x2+a, one can follow these steps Maximization and minimization: A point on a smooth function where the derivative is zero is a local maximum, a local minimum, or an inflection point of the function. This can be clearly seen in the figure, where the function has three points at which the tangent to the curve is horizontal (the slope is zero). This function has a local maximum at A, a local minimum at B, and an inflection point at C. One can determine whether a point with zero derivative is a local maximum, a local minimum, or an inflection point by evaluating the value of the second derivative at that point. Page 4 of 23 Given a smooth function f(x), one can find the local maximums, local minimums, and inflections points by solving the equation to get all points x that have a derivative of zero. One can then check the second derivative for each point to get the specific character of the function at each point. Global maximum and minimum: The global maximum or minimum of a smooth function in a specific interval of its argument occurs either at the limits of the interval or at a point inside the interval where the function has a derivative of zero. As can be seen in the figure, the function shown has a global maximum at point A on the left boundary of the interval under consideration, a global minimum at B, a local maximum at C, and a local minimum at D. Partial derivative: Page 5 of 23 The derivative of a function of several variable with respect to only one of its variables is called a partial derivative. Given a function f(x,y), its partial derivative with respect to its first argument is denoted by and defined by Since all other variables are kept constant during the partial derivative, it represents the slope of the curve one obtains when varying only the designated argument of the function. For example, the partial derivative of the function with respect to x is evaluated by treating y as a constant so that one gets The chain rule: Consider a function f[U(x), V(x)] of two arguments U and V, each a function of x. The chain rule can be used to find the derivative of f with respect to x by the rule For example, consider the function f = UV2 where U =2x+3 and V=4x2. The derivative of f with respect to x is given by Integration: Page 6 of 23 The integral of a function f(x) over an interval from x1 to x2 yields the area under the curve of the function over this same interval. Let F denote the integral of f(x) over the interval from x1 to x2. This is written as and is called a definite integral since the limits of integration are prescribed. The area under the curve in the following figure can be approximated by adding together the vertical strips of area . Therefore, the integral is approximated by This approximation approaches the value of the integral as the width of the strips approaches zero. Indefinite Integrals: Page 7 of 23 A function F(x) is the indefinite integral of the function f(x) if The indefinite integral is also know as the anti-derivative. Since the derivative of a constant is zero, the indefinite integral of a function can only be evaluated up to the addition of a constant. Therefore, given a function F(x) to be an anti-derivative of f(x), the function F(x) + C, where C is any constant, is also an anti-derivative of f(x). This constant is known as the constant of integration and may be determined only if one has additional information about the integral. Normally, a known value of the integral at a specified point is used to calculate the constant of integration. The basic rules of integration are The indefinite integral of commonly used functions: The following is a list of indefinite integrals of commonly used functions, up to a constant of integration [ ]: Page 8 of 23 Note: Remember to add a constant of integration. You evaluate the constant of integration by selecting the constant of integration such that the integral passes through a known point. Relating definite and indefinite integrals: To obtain the value of the definite integral knowing the value of the indefinite integral of the function, one can subtract the value of the definite integral evaluated at the lower limit of integration from its value at the upper limit of integration. For example, if you have the indefinite integral Note that C, the constant of integration, cancels in the subtraction and need not be included. It is common to sometimes use the notation Change of variables: Given a function U(x), one can use to change the variable of integration from x to U . The change of variables results in the rule Page 9 of 23 For example, given the function we can write this function as where U = ax+b and . Therefore, the integral of f can be evaluated by using the following steps. Change of variables for a definite integral is similar with an additional change in the limits of integration. The resulting equation is For example, given the function , we can write this function as where U = ax and . Therefore, the integral of f can be evaluated by using the following steps. Integration by parts: Given the functions U(x) and V(x), one can use integration by parts to integrate the following integral using the relation Page 10 of 23 For example, to evaluate the integral one can take and so that and . Using integration by parts we get Integration over a line: The integral F of function f(s) over line AB, that is defined by s = 0 to s = l, is written as . When either the domain of integration or the function is described in terms of another variable, such as x in the figure, one can evaluate F by a change of variables to get Depending on the format the information is provided in, it might be necessary to use the Pythagorean Theorem to relate the differential line element along the arc of the curve to the x and y coordinates. For example, in the figure shown we can see that Page 11 of 23 The sign of the root must be selected based on the specifics of the problem under consideration. For example, consider integrating the function f(x,y) =xy2 over the straight line defined in the figure from point A to point B. Direct integration, using the relations would yield Page 12 of 23 Double integral: The double integral of function f(x,y) first integrating over x and then integrating over y is given by The notation implies that the inner integral over x is done first, treating y as a constant. Once the inner integral is completed and the limits of integration for x are substituted into the expression, the outer integral is evaluated and the limits for y are substituted into the resulting expression. The rules of integration are the same as used for single integration for both the integration over x and the integration over y. To integrate over y first and then over x, the integration would be written as Page 13 of 23 One can also write the integral limits without specifying the variable (i.e., without using "x=" and "y="). The order dxdy or dydx clearly specifies what variable a specific limit is associated with. Consider the following example of double integration of the function f(x,y) =xy2. Unlike the example, the limits of integration need not be constants.