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Section 4.1 Section 4.2 Section 4.3 Applications of Derivatives Calculus for the Biological Sciences February 24, 2006 Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Definition: A function f (x) is said to be an increasing function over a certain interval of values of x if f (x) increases with increases of x. That is, if x1 and x2 are any two values in the given interval with x2 > x1 , then f (x2 ) > f (x1 ). Definition: A function f (x) is said to be a decreasing function over a certain interval of values of x if f (x) decreases with increases of x. That is, if x1 and x2 are any two values in the given interval with x2 > x1 , then f (x2 ) < f (x1 ). Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Theorem 4.1.1: (a) If f (x) is an increasing function that is differentiable, then f 0 (x) ≥ 0. (b) If f (x) is a decreasing function that is differentiable, then f 0 (x) ≤ 0. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Theorem 4.1.2: (a) If f 0 (x) > 0 for all x in some interval, then f (x) is an increasing function of x over that interval. (b) If f 0 (x) < 0 for all x in some interval, then f (x) is a decreasing function of x over that interval. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Example: Find the values of x for which the function f (x) = x 2 − 2x + 1 is increasing or decreasing. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Definition: A function f (x) is said to be convex(concave upwards) 00 over a certain interval of values of x if f > 0. Definition: A function f (x) is said to be concave(concave 00 downwards) over a certain interval of values of x if f (x) ≤ 0. A point of inflection on a curve is a point where the curve changes its convexity. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Example: Find the values of x for which the function 1 f (x) = x 4 − x 3 + 2x 2 6 is concave upwards or concave downwards.increasing or decreasing. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Definition: A point of inflection on a curve is a point where the curve changes from concave upwards to concave downwards, or vice versa. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Example: Find the points of inflection of y = x 1/3 . Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Theorem 4.1.3 (Rolle’s Theorem): Let the function f (x) be defined and continuous at all x in the closed interval a ≤ x ≤ b and be differentiable at all x in the open interval a < x < b. Let f (a) = f (b). There exists at least one number c satisfying a < c < b at which f 0 (c) = 0. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Analysis of Curves Theorem 4.1.4 (Mean Value Theorem): Let f (x) be continuous on the closed interval a ≤ x ≤ b and differentiable on the open interval a < x < b. Then there exists at least one number c satisfying a < c < b such that f 0 (c) = f (b) − f (a) . b−a Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Definition: i. A function f (x) is said to have a local maximum at x = c if f (c) > f (x) for all x sufficiently near c. ii. A function f (x) is said to have a local minimum at x = c if f (c) < f (x) for all x sufficiently close to c. iii. The term extremum is used to denote either local maximum or a local minimum. iv. The value x = c is called a critical value for a continuous function f if 1. either f 0 (c) = 0 or f 0 (x) fails to exist at x = c and if 2. f (c) is well-defined. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Theorem 4.2.1 (Second Derivative Test): Let f (x) be twice differentiable at x = c. Then 1. x = c is a local maximum of f whenever f 0 (c) = 0, f 00 (c) < 0; 2. x = c is a local minimum of f whenever f 0 (c) = 0, f 00 (c) > 0. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Example: Determine the local maximum and minimum values of x 3 + 2x 2 − 4x − 8. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Theorem 4.2.2 (First Derivative Test): If x = c is a critical value for f (x), that is, either f 0 (c) = 0 or f 0 (x) fails to exist as x → c: 1. x = c is a local maximum of f if f 0 (x) changes sign from positive to negative as x changes from just below c to just above c. 2. x = c is a local minimum of f if f 0 (x) changes sign from negative to positive as x changes from just below c to just above c. 3. x = c is not a local extremum if f 0 (x) does not change sign as x changes from just below c to just above c. In this case x = c will be either a point of inflection or a corner on the graph of f (x). Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Example: Find the local extrema for f (x) = Calculus for the Biological Sciences x4 . (x − 1) Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Definition: The absolute maximum and absolute minimum is the largest and smallest values of f (x) respectively taken over the entire domain. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Example: Determine the absolute maximum and minimum values of f (x) = 1 + 12x − x 3 over the interval 1 ≤ x ≤ 3. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Maxima and Minima Example: Find the absolute maximum and minimum values of f (x) = x2 x2 + 1 on the interval [−1, ∞). Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Applications of Maxima and Minima Example: Find two numbers whose sum is 16 and whose product is the largest possible. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Applications of Maxima and Minima Example: Construct a tank with a square base, rectangular sides and no top. The tank must hold 4m3 of water. The material used to make the tank costs $10/m2 . Determine the dimensions of the tank which will minimize the cost. Calculus for the Biological Sciences Applications of Derivatives Section 4.1 Section 4.2 Section 4.3 Applications of Maxima and Minima Example: A grain silo is to be built in the form of a vertical cylinder with a hemispherical roof (Fig. 4.34 page 189). The silo is to be capable of storing 10, 000ft 3 of grain. No grain is stored in the roof. The hemispherical roof costs twice as much per unit area to manufacture as the cylindrical sides cost. What dimensions of the silo will minimize the cost? Calculus for the Biological Sciences Applications of Derivatives