Calculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is increasing on an interval if x ≤ y implies f(x) ≤ f(y), and decreasing if x ≤ y implies f(x) ≥ f(y). It is called monotonic if it is either increasing or decreasing on its entire domain. Example 1.1 ex is increasing on (−∞; 1) and ln x is increasing on (0; 1). f(x) = 1=x is decreasing on (−∞; 0) and on (0; 1). The sine function is increasing on [−π=2; π=2], [3π=2; 5π=2], etc. A function f(x) is proportional to another function g(x) if there is a nonzero constant k such that f(x) = kg(x). Example 1.2 The force of gravity F between two masses M and m is proportional to the inverse square function of the distance between them, r, namely GMm F = r2 2 Here k = GMm and g(r) = 1=r . The graph of a function f is concave up if it bends upward as we move from left to right, that is if, when we pick two points (x1; y1) and (x2; y2) on the graph of f, the graph lies below or on the line segment joining those points on that inteval. Similarly, the graph of f is concave down if it bends downward, or if the graph of f lies above or on any line segment joining two points on its graph. An inflection point is a point (x; f(x)) on the graph of f at which the graph changes concavity (or equivalently just the x value of that point). π 3π π π Example 1.3 The cosine function is concave up on [ 2 ; 2 ] and concave down on [− 2 ; 2 ], etc. It has inflection points at odd multiples of π=2. A function f is said to be even if f(−x) = f(x) and odd if f(−x) = −f(x). Even functions are symmetric about the y-axis, while odd ones are symmetric about the origin. 2 Example 1.4 The cosine function is even, f(x) = jxj is even, and f(x) = e3x is even, 3 while the sine function is odd, f(x) = x is odd, and f(x) = arctan(x) is odd. 1.2 Limits and Continuity Suppose f is defined on an interval around a given real number c, except perhaps at x = c itself. The limit of f as x approaches a given real number c is a real number L satisfying the following condition: for all > 0 there is a δ > 0 such that jx − cj < δ and x 6= c 1 implies jf(x) − Lj < In this case we write lim f(x) = L x!c Remark 1.5 In plain English, this means that the limit L exists provided the following holds: however small I want to make my -window on the y-axis around L, I can find a small enough δ-window on the x-axis around c which is mapped by f entirely into the -window. Remark 1.6 Note the logical implications: I have freedom to choose . I can choose it to be one in a billion, whatever I want. But once chosen, I'm typically restricted in my choice of δ. For example, if f(x) = x2 and L = 100, then clearly c = 10, and to show that the limit as x approaches 10 is 100, I need to find a δ for any given . Suppose I've chosen my , say = 1=1000. Then you can check that a good enough δ is δ = 1=21000. It's not the only one, since I can pick it to be smaller than this, but I am restricted by how big I can make it. It can't be much bigger than that. Remark 1.7 Note also that the definition of limit includes absolute values around x − c. We write jx − cj and this implies the existence and equality of the left and right limits, lim f(x) and lim f(x) x!c− x!c+ respectively. x−1 Example 1.8 The function f(x) = jx−1j has left and right limits as x approaches 1, but the two limits are not equal. Hence f has no limit at x = 1. We can analogously define lim f(x) x→±∞ The difference is we can't use the δ part of the definition. The part is the same, we can choose any > 0 such that jf(x)−Lj < under some appropriate condition on x. It's just that condition can't be the existence of δ such that jx − cj < δ (because c = 1 and jx − 1j = 1, which is never less than δ), so instead we demand the existence of an M > 0 such that x ≥ M implies jf(x) − Lj < δ. A function f is continuous at a point x = c if the limit limx!c f(x) = L exists and moreover L = f(c), i.e. if lim f(x) = f(c) x!c We say f is continuous on an interval [a; b] or (a; b), etc., if it is continuous at all points in that interval. Example 1.9 All polynomials are continuous on (−∞; 1). All rational functions p(x)=q(x) 2 are continuous except at zeros of q(x). For example, 2x +3 is continuous everywhere except p x2−5 at ± 5. The exponential function ex is continuous on (−∞; 1), the log function ln x is continuous on (0; 1). The sine and consine functions are continuous on (−∞; 1). The absolute value function f(x) = jx − 2j is continuous everywhere on (−∞; 1). 2 Discontinuities, or points of discontinuity of f, are x-values where f is not continuous. For example 0 is a point of discontinuity of csc x and of f(x) = 1=x.A removable disconti- x2−4 nuity is one that can be \plugged", e.g. x = −2 is a removable discontinuity of f(x) = x+2 . An essential discontinuity is one that cannot be \plugged", typically because f goes to ±∞ near that point, for example x = 1 is an essential discontinuity of f(x) = 2=(x − 1). A jump discontinuity is a discontinuity where f \jumps" a finite amount near it, for example x−1 x = 1 for f(x) = jx−1j . 1.3 Rate of Change and the Derivative The average rate of change of a function f between x = a and x = b is defined to be the slope of the line connecting the points (a; f(a)) and (b; f(b)) on the graph of f: f(b) − f(a) average rate of change = b − a If we define h = b − a, then b = a + h, and the average rate of change between a and b = a + h becomes the difference quotient: f(a + h) − f(a) h If the limit of the difference quotient as h approaches 0 exists, then we say that f is differen- tiable at x = a and we call this limit the derivative of f at a. It is a real number, denoted equivalently by 0 df dy f(a + h) − f(a) f (a) ≡ (a) ≡ = lim dx dx x=a h!0 h If f is differentiable on an entire interval, then varying the point a, in other words letting x vary, we get a function, the derivative function f 0(x). One of our tasks is to find various derivative functions for well-known functions. Example 1.10 Common derivative functions are the following: 1. (xn)0 = nxn−1 (power rule) 2. (sin x)0 = cos x 3. (cos x)0 = − sin x 4. (tan x)0 = sec2 x 5. (ex)0 = ex 0 1 6. (ln x) = x 7. (ax) = (ln a)ax 0 1 8. (arctan x) = 1+x2 9. (arcsin x)0 = p 1 1−x2 3 2 Remark 1.11 To compute more complicated derivatives, such as those of xex , for example, we will need to know how to break up the process of taking a derivative into several steps involving only derivatives of things we know how to compute, such as (1)-(6) above. For example, we know how to take the derivative of x, x2 and ex, and we will develop methods 2 to compute (xex )0 in terms of these easier ones. This is the content of the sum, product, quotient and chain rules below. The derivative f 0(a) at x = a can be interpreted as the slope of the tangent line to the graph of f at the point (a; f(a)). Since we have a slope, f 0(a), and a point, (a; f(a)), we can find the equation of the tangent line using point-slope: y − f(a) = f 0(a)(x − a) Adding f(a) to both sides gives the equation for the tangent line, which we also call the linear approximation to f near x = a or local linearization near x = a: y = f(a) + f 0(a)(x − a) This is the key idea behind the idea of a derivative, to approximate a complicated function f(x) with the simplest one possible, a line, at least locally. The tangent line is an honest-to- f(x)−f(a) 0 goodness approximation of f near x = a, because we know that as x ! a, x−a ! f (a), i.e. from the definition of limit we get f(x) − f(a) ≈ f 0(a) () f(x) − f(a) ≈ f 0(a)(x − a) multiply both sides by (x − a) x − a () f(x) ≈ f(a) + f 0(a)(x − a) add f(a) to both sides Example 1.12 Let us approximate the value of f(x) = esin x near x = π using local linearization, say at x = 3 which is near x = π.