Rational functions and average rates of change Math 102 Section 106

Transcription

1 Rational functions and average rates of change Math 102 Section 106 Cole Zmurchok September 12, 2016

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4 Last time: power functions From Figure 1.1a, we see that the power functions (y = x n for po intersect at x =0and x =1. This is true for all integer powers. The demonstrates another extremely important fact: the greater thepower graph near the origin and the steeper the graph beyond x>1. Thiscanb of the relative size of the power functions. We say that close to the or with lower powers dominate, while far from the origin, the higher power Small powers dominate close to x = 0; large powers dominate for large x. y y x 5 x 4 x 3 x 2

5 Last time: power functions and curve sketching Example (Sketch y = x 5 + ax 3.) a < 0: a = 0: a > 0:

6 Rational functions A rational function is a function that can be written as y = p 1(x) p 2 (x), where p 1 (x) and p 2 (x) are polynomials. Example (Hill function) Draw a sketch of for x 0. y = Axn a n + x n

7 Rational functions Helpful notation: y = Axn a n + xn, x 0. x a means x much smaller than a. x a means x much bigger than a.

8 Rational functions y = Axn a n + xn, x 0. x a (think1.5. small Rate x): of an enzyme-catalyzed reaction a n + x n a n y Axn a = A. Small x n a nxn Large x y y n=1 A 2 3 x

9 Rational functions y = Axn a n + xn, x 0. x a (think big x):.5. Rate of an enzyme-catalyzed reaction a n + x n x n y Axn = A x n Small x Large x Smoothly connecte y y A y n=1 2 3 n=1 n=2 x x

10 Hill function: y = Axn a n +x n, x Rate of an enzyme-catalyzed reaction 13 Small x Large x Smoothly connected y y n=3 A y n=1 2 3 n=1 n=2 x x x Figure 1.5. The rational functions (1.7) with n =1, 2, 3 are compared on this graph. Close to the origin, the function behaves like a power function, whereas for large x there is a horizontal asymptote at y = A. As n increases, the graph becomes flatter close to the origin, and steeper in its rise to the asymptote Saturation and Michaelis-Menten kinetics

11 Hill function: y = Axn a n +x n, x 0 Q1. The asymptote for the Hill function is A. A B. A/2 C. a D. a/2 E. a n x y y = A is the maximal response We also say lim x Ax n a n + x n = A.

12 Hill function: y = Axn a n +x n, x 0 Q2. The value of x for half-maximal response is A. A B. A/2 C. a D. a/2 E. a n x y x = a is the half-max response

13 Hill function: y = Axn a n +x n, x 0 Q3. Why is it called a Hill function? A. Because it looks like a hill B. Because it describes an increasing function C. Because it was named after A.V. Hill D. Because in biology it describes a Hill process

14 Why is it called a Hill function? Hill functions are named after, Archibald Hill, a Nobel Prize winning muscle physiologist. The Combinations of Haemoglobin with Oxygen and with Carbon Monoxide. I Biochem. J 1913 Oct; 7(5): articles/pmc / wikipedia.org/ wiki/ Archibald_Hill

15 Speed of an enzyme reaction Michaelis-Menten kinetics: E + S C E + P speed of reaction = v = Kx k n +x

16 Speed of a cooperative enzyme reaction E + 2S C E + P w speed of reaction = v = Axn a n +x n Hill functions:

17 Average rate of change Suppose that y = f(t) is a function. The average rate of change of f over the interval a t b is Change in f Change in t = f f(b) f(a) = t b a 2.3. The slope of a secant line is the average rate of change f(b) y=f(t) f(x o+h ) secant line secant lin f(a) f(x ) o a b t x

18 Average rate of change Q4. Let y = f(x) be a function. The average rate of change of f over the interval y=f(t) x 0 x f(b) x 0 + h is f(x A. 0 ) f(h) secant line B. C. D. E The slope of a secant line is the average rate of change 29 x f(x 0 f(a) +h) f(h) h f(x 0 +h) f(x 0 ) a x 0 b t f(x o+h ) f(x ) o secant line x +h f(x 0 +h) f(x 0 ) Figure x 2.4. Asecantlineisastraightlineconnectingtwopointsonthegraph of f(x a function. 0 +h) f(x Left: a set 0 ) of time dependent data points (black circles)orsmoothfunction (dashed curve) f(t) showing the secant line through the points (a, f(a)), and(b, f(b)). Right: The graph h of some arbitrary function f(x) with a secant line through the points (x 0,f(x 0 )) and (x 0 + h, f(x 0 + h)). The slope of the secant line is the same as as the average rate of change of f over the given interval. x o y=f(x) o x

19 Average rate of change The average rate of change of y = f(x) over the interval y=f(t) f(b) x 0 x x 0 + h is 2.3. The slope of a secant line is the average rate of change 29 secant line y x = f(x f(a) 0 + h) f(x 0 ) (x 0 + h) x 0 = f(x a b 0 + h) f(x 0 ) h t f(x o+h ) f(x ) o secant line x o y=f(x) x +h Figure 2.4. Asecantlineisastraightlineconnectingtwopointsonthegraph of a function. Left: a set of time dependent data points (black circles)orsmoothfunction (dashed curve) f(t) showing the secant line through the points (a, f(a)), and(b, f(b)). Right: The graph of some arbitrary function f(x) with a secant line through the points (x 0,f(x 0 )) and (x 0 + h, f(x 0 + h)). The slope of the secant line is the same as as the average rate of change of f over the given interval. o x

20 Zebrafish development Posterior lateral line primordium migration:

21 Zebrafish development Figure modified from Valdivia et al. Development Q5. Over time the cell cluster is A. speeding up B. slowing down C. moving at the same speed

22 Average velocity Suppose the cell cluster is at position y 1 at t 1 and at y 2 at t 2. The average velocity over the interval t 1 t t 2 is distance traveled v average = time taken = y t = y 2 y 1 t 2 t 1 The average velocity depends on the time interval considered!

23 Zebrafish development time t (hours) position y (µm) Average velocity spreadsheet

24 Zebrafish development time t (hours) position y (µm) Q6. Over 2 t 4 hours, the average velocity of the cell cluster was A. 150 µm/h B. 100 µm/h C. 75 µm/h D. 50 µm/h E. 25 µm/h

25 Secant line Q7. A secant line is A. A line whose slope is instantaneous velocity B. A line connecting two points on a graph C. The same as average velocity D. The same all along the curve E. Not sure

26 Zebrafish development 400 y t Blue curve: position as a function of time Black dots: measured data points Red line: secant line through (t 1, y 1 ) and (t 2, y 2 )

27 Zebrafish development 400 y t Blue curve: position as a function of time Black dots: measured data points Red line: secant line through (t 1, y 1 ) and (t 2, y 2 )

28 Zebrafish development 400 y t v average = y t = y 2 y 1 t 2 t 1 which is the slope of the secant line through (t 1, y 1 ) and (t 2, y 2 ).

29 Today... Rational functions Hill functions (horizontal asymptote, half-max) Enzyme reaction speed can be modelled using Hill functions Average rate of a function Average velocity of cluster of cells Secant lines

30 Answers 1. A 2. C 3. C 4. E 5. B 6. C 7. B

31 Related exam problems 1. Consider the position of a particle (y) as a function of time (t), given by the formula 2. y = f(t) = k 1 t 3 k 2 t. Find the average velocity of the particle over the time interval 1 t 2.