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Chapter 2 Calculus: Hughes-Hallett The Derivative Continuity of y = f(x) A function is said to be continuous if there are no “breaks” in its graph. A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a. Continuous Functions- The function f is continuous at x = c if f is defined at x = c and lim f ( x) f (c). xc The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. If f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.) Definition of Limit- Suppose a function f, is defined on an interval around c, except perhaps not at the point x = c. The limit of f(x) as x approaches c:lim f ( x ) L x c is the number L (if it exists) such that f(x) is as close to L as we please whenever x is suffici- ently close to c (but x c). In lim f ( x ) L , 0, 0, x c Symbols: , , 0, 0, if 0 | x c | then | f ( x ) L | . Properties of Limits- Assuming all the limits on the right hand side exist: 1. lim bf ( x) b lim f ( x) , b a constant x c x c 2.lim f ( x) g ( x) lim f ( x) lim g ( x) x c x c x c 3.lim f ( x) g ( x) (lim f ( x))(lim g ( x)) x c x c x c f ( x) lim f ( x) 4.lim x c , provided lim g ( x) 0 x c g ( x ) lim g ( x) x c x c 5. lim b b 6. lim x c x c x c Limits at Infinity- If f(x) gets as close to a number L as we please when x gets sufficiently large, then we write: lim f ( x) L x Similarly, if f(x) approaches L as x gets more and more negative, then we write: lim f ( x) L x Average Rate of Change- The average rate of change is the slope of the secant line to two points on the graph of the function. y 2 y1 f ( x 2 ) f ( x1 ) f ( x1 h ) f ( x1 ) m sec x 2 x1 x 2 x1 h The Derivative is -- Physically- an instantaneous rate of change. Geometrically- the slope of the tangent line to the graph of the curve of the function at a point. Algebraically- the limit of the difference quotient as h 0 (if that exists!). In symbols: dy df f(x1 h ) f ( x1 ) f(x1 x ) f ( x1 ) f ' ( x1 ) lim lim dx dx h 0 h x 0 x First Derivative Interpretation- If f’ > 0 on an interval, then f is increasing over the interval. If f’ < 0 on an interval, then f is decreas- ing over the interval. Derivative Symbols: If y = f(x) = x 3 3x 3 then each of the following symbols have the same meaning: dy df df ( x ) d( x 3 3x 3) f ' ( x ) y' D x f D x y 3x 2 3 dx dx dx dx And at a particular point, say x = 2, these symbols are used: dy f ' (2) y' (2) D x f (2) 3(2) 2 3 9 dx x 2 Basic Formulas (1): Derivative of a constant: If f(x) = k, the f’(x) = 0, k - a constant Derivative of a linear function: If f(x) = mx + b, then f’(x) = m Derivative of x to a power: If f ( x ) x n , then f ' ( x ) nx n 1 Second Derivative Interpretation- If f’’ > 0 on an interval, then f’ is increasing, so the graph of f is concave up there. If f’’ < 0 on an interval, then f’ is decreasing, so the graph of f is concave down there. If y = s(t) is the position of an object at time t, then: Velocity: v(t) = dy/dt = s’(t) = s ( t ) Acceleration: a(t) =d y / dt s' ' ( t ) v' ( t ) s v 2 2 Continuous Functions- The function f is continuous at x = c if f is defined at x = c and lim f ( x) f (c). xc The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. If f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.) Continuity of y = f(x) A function is said to be continuous if there are no “breaks” in its graph. A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a. Theorem on Continuity- Suppose that f and g are continuous on an interval and that b is a constant. Then, on that same interval: 1. bf(x) is continuous. 2. f(x) + g(x) is continuous. 3. f(x)g(x) is continuous. 4. f(x)/g(x) is continuous, provided g( x) 0 ` on the interval. Differentiability- A function f is said to be differentiable at x = a if f’(a) exists. Theorem: If f(x) is differentiable at a point x = a, then f(x) is continuous at x = a. Linear Tangent Line Approximation- Suppose f is differentiable at x = a. Then, for values of x near a, the tangent line approximation to f(x) is: f ( x) f (a) f ' (a)(x a). The expression f (a) f ' (a)(x a) is called the local linearization of f near x = a. We are thinking of a as fixed, so that f(a) and f’(a) are constant. The error E(x), is defined by E ( x) f ( x) f (a) f ' (a)(x a). and lim E ( x) 0. x a xa

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