CHM320: MATH REVIEW. I. Ordinary Derivative:

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1 CHM30: MATH REVIEW I. Ordinary Derivative: Figure : Secant line between two points on a function Ordinary derivatives describe how functions of a single variable change in response to variation of the independent variable. Consider a function f in Fig. (green curve in color) where the x axis represents the values of independent variable x and the y axis represents the values of function f at each value of x. To define the ordinary derivative of function f at x, which is usually denoted as f (x), consider two points on the x axis: x and x+h (see Fig. ). x is a point on the x-axis where we want to calculate the derivative of function f. h is a small number. The values of function f at these two points are f(x) and f(x+h), respectively. The straight line through two points, (x,f(x)) and (x + h,f(x + h)), is often called secant line (purple line in Fig. ). To understand the meaning of derivative, we first need to define the slope of secant line shown in Fig., which is given by slope = f x = f(x+h) f(x) h where x = (x + h) x = h is the difference between two points on the x-axis and f = f(x + h) f(x) is the difference in the values of function f at x and x + h. Now, let s see what happens to the secant line and, more importantly, its slope as h decreases. As shown in Fig. (next page), the secant line becomes the tangent line at x as h approaches zero (i.e. h h h 0). Accordingly, the slope of secant line approaches the slope of the tangent line at x. The slope of the tangent line attached to the function f at x is the (first) derivative of f at x, or f (x). Mathematically, the derivative of f at x is defined as () f (x) = df(x) dx = lim f(x+h) f(x) h 0 h where another notation for derivative, df/dx, is also introduced. Notice the similarity in notation ()

2 Figure : A series of secant lines (left) and the tangent line (right) between f/ x and df/dx. Roughly speaking, df is the (infinitesimal) change in the value of f in response to an infinitesimal change in x, dx. Therefore, the ratio between df and dx has the same physical meaning as the slope. In summary, f (x) is a function representing the slope of the tangent line attached to f at x (Fig. ), which shows how fast the function changes upon given amount of change in x. Derivative of some important functions Important properties of derivative f(x) f (x) f(x) f (x) x n nx n e x e x sinx cosx lnx /x cosx -sinx f(x) = g(x)+h(x) f (x) = g (x)+h (x) (3) f(x) = g(x) h(x) f (x) = g (x) h(x)+g(x) h (x) (4) f(x) = g(x) h(x) f (x) = g (x)h(x) g(x)h (x) (5) [h(x)] f(x) = g(h(x)) f (x) = g (h(x))h (x) (6) Examples example for (3): f(x) = x 4x+5 f (x) = x 4 This means that the slope of tangent line attached to f at x = is f () = 4 =.

Definite Integral A definite integral of a function gives us the area under the curve.

3 example for (4): f(x) = x sinx f (x) = xsinx+x cosx examples for (5): examples for (6): f(x) = sinx x f (x) = xcosx sinx x f(x) = lnx x f (x) = lnx x f(x) = sinx f (x) = x(cosx ) f(x) = e 3x f (x) = (e 3x )6x II. Integral: A. Definite Integral A definite integral of a function gives us the area under the curve. To understand this interpretation of definite integral, let s take a look at a function f(x) shown in Fig. 3. The definite integral of function f(x) between two points, x = a and x = b (denoted as b a f(x)dx), is the sum of Figure 3: Definite integral between x = a and b Figure 4: Riemann sum 3

4 the areas of three shaded regions. Notice that, if the function is positive, the area is also positive, but, if the function is negative, the area becomes a negative quantity. Therefore, the middle area in Fig. 3 (shaded yellow in color) has a negative contribution to the definite integral of f(x) between x = a and b. A practical way to compute the area under the curve is to approximate the area by a collection of rectangles. First, let s assume that we divide the whole range, [a, b], into N equally spaced intervals. Therefore, each interval has the lengh of x = (b a)/n and we label the starting point of j th interval as x j. In other word, x = a,x = a+ x,x 3 = a+ x,.. and so on. Then, we place N rectangles between x = a and b, all of which have base of size x. The height of j th rectangle is equal to f(x j ). Such arrangement of rectangles is schematically shown in Fig. 4. Closed circles in Fig. 4 represent the starting point of each rectangle (i.e. x j s). As you can see, the sum of all the areas represented by rectangles will be a good approximation to the area under the curve. This approximate area represented by rectangles is often called Riemann sum, which is given by N Area (rectangles) = f(x j ) x (7) j= where f(x j ) x is the area of j th rectangle (height base). It is apparent from Fig. 4 that the Riemann sum becomes closer to the area under the curve as the number of rectangles increases (i.e. the bases of rectangles decrease, or x 0) and, eventually, become identical to the area under the curve with infinitely many rectangles. Therefore, the area under the curve can be represented by the limit of Riemann sum as ( ) N b a b Area (under the curve) = lim f(x i ) = f(x)dx (8) N i= N a where (b a)/n in Eq. (8) is x in Eq. (7). The infinite summation in Eq. (8) can be regarded as the definite integral of f(x) between x = a and b. B. Indefinite Integral Now that we understand the meaning of definite integral, the next thing to do is to figure out an efficient way to compute the definite integral. One way to accomplish this is to follow Eq. (8): compute the Riemann sum and find out the limiting value of the sum as N. However, this is awfully inconvenient and, often, very difficult to carry out. It turns out there is a much easier way to carry out the definite integral of f(x). First, we need to find out the indefinite integral of f(x). A function g(x) is called the indefinite integral of f(x) if the derivative of g(x) is the given function 4

5 f(x). In other word, f(x)dx = g(x) if g (x) = f(x) (9) Therefore, in some sense, indefinite integral is the opposite of derivative. Note that the result of indefite integral of a function is another function whereas the result of definite integral is a number. The definite integral of f(x) between x = a and x = b is then given by b a f(x)dx = g(x) b a = g(b) g(a) (0) This implies that the values of indefinite integral, g(x), at the end points of definite integral (i.e. x = a and b) are all that we need to compute the definite integral. Indefinte integral of some important functions f(x) f(x) f(x) f(x) Important properties of integral x n (n+) xn+ e x e x sinx -cosx /x lnx cosx sinx lnx xlnx x (f(x)+g(x))dx = f(x)dx+ g(x)dx () b a f(x)dx = f(x)dx () a b cf(x)dx = c f(x)dx (3) c a f(x)dx = b a f(x)dx+ c b f(x)dx, b [a,c] (4) Examples π 0 sinxdx = cosx π 0 = cosπ +cos0 = + = 5 ( x + 3 ) ( ) 5 dx = x 3 x3 +3lnx = 3 (53 3 )+3(ln5 ln) = 78+3ln(5/) 5

6 EXERCISES. Compute the ordinary derivatives of following functions: (a) cos3x+sin3x (b) sinx (c) x lnx (d) lnx x (d) e x3 lnx. Compute the following definite integrals. (a) x dx (b) e x/3 dx (c) x dx 6

7 MATH EXERCISE (Keys). Compute the ordinary derivatives of following functions: (a) f(x) = cos3x+sin3x f (x) = 3sin3x+6cos3x (b) f(x) = sinx f (x) = (4x)cosx (c) f(x) = x lnx f (x) = (4x)lnx +4x (d) f(x) = lnx x f (x) = xx +lnx( x ) = ( lnx)/x 3 3 (d) f(x) = e x3 lnx f (x) = e x3 3x lnx +e x3 x x = e x3 (3x lnx +/x). Compute the following definite integrals. (a) x dx x dx = lnx = (ln) (ln) = ln (b) e x/3 dx ex/3 dx = 3e x/3 7

8 = 3e /3 3e /3 (c) x dx x dx = [ x = ] [ ] = 8

MATH 116, LECTURE 13, 14 & 15: Derivatives

MATH 116, LECTURE 13, 14 & 15: Derivatives 1 Formal Definition of the Derivative We have seen plenty of limits so far, but very few applications. In particular, we have seen very few functions for which