alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Exer ercise ex er cise 3 divide 10 into two par ts

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Unformatted text preview: ti v e (>0). increasing positiv Conversely : if y ’(x) is positi v e then y(x) is incr easing positiv increasing easing. minimum. Let f(x) be a function and b the point where f(x) is minimum At a minimum point b the value of the function is lesser than the values around it. Let δ x be a small change in x. b b Then f(b -- δ x) > f(b) and f(b + δ x) > f(b) b b At a minimum point f ’(x) is neg tiv befor zer positiv after. ne g a ti v e bef or e , z er o , positi v e after What can you say about f ”(b) ? b 114 Exercise 1: Find an instant where f(x) = x 2 + x h as an extreme value (maximum or minimum). Exercise 2: Find an instant where f(x) = -- x 2 + x h as an extreme value (maximum or minimum). Exer ercise Ex er cise 3: Draw the curves of the functions below and answer the following : a) f(x) = x 2 for x < 0 . The shape of the curve is concave upwards and falling. b) f(x) = -- x 2 f or x > 0 . The shape of the cur ve is concave downwards and falling. c) f(x) = x 2 for x > 0 . The shape of the curve is concave upwards and rising. d) f(x) = -- x 2 for x < 0 . The shape of the cur ve concave downwards and rising. i) What kind of angle does the tangent to the cur ve make ? (acute, obtuse) . What is the sign of the tangent ? ii) What is the sign of the first derivative of the function ? (positive, negative) iii) Is the function increasing or decreasing ? Exercise 4: Repeat exercise 3 for the functions below and describe the shape of the curve. a) f(x) = x 3 f or x < 0 . b) f(x) = x 3 f or x > 0 . c) f(x) = -- x 3 f or x < 0 . d) f(x) = -- x 3 f or x > 0 . 115 2 3. MAXIMA and MINIMA Given a function f(x) how can we find the instants where f(x) has maximum or minimum values ? If the point a is an extr eme then the first derivative must equal zero. xtreme Solve the equation f ’(a) = 0 to find a . a This information is necessary but not sufficient to know whether the point a is a maximum point or a minimum point. There are three methods. Method 1 : We need to fur ther check around a if...
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