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2.4 Rates of change and tangent lines

2.4 Rates of change and tangent lines. Objectives: To find average rate of change of a function. Average rate of change. The average rate of change of f from x 1 to x 2 is if x 1  x 2 . This is also called the difference quotient . Another version is

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2.4 Rates of change and tangent lines

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  1. 2.4 Rates of change and tangent lines Objectives: To find average rate of change of a function

  2. Average rate of change • The average rate of change of f from x1 to x2 is if x1  x2. This is also called the difference quotient. Another version is where h is the amount of change in x

  3. A secant line is a line that touches the graph at two points. The average rate of change of a function equals the slope of the secant line. y = f(x) Secant Line (x2, f(x2)) f(x2) - f(x1) (x1, f(x1)) x2 – x1

  4. Numerical Example: If 0.09 grams of E.coli is placed in a petri dish and allowed to grow the results are shown on the following table • Find the average rate of change for each time interval. What is happening to the rate of change as time passes? What does the rate of change represent?

  5. Example: Graphically • Use the graph of f(x) = x2 to find the rate of change from 0 to 2 and from 2 to 3

  6. Example: Analytically 1. Find the average rate of change of f(x) = 2x2 – 3x from 1 to 2 2. What is the slope of the secant line containing (1,f(1)) and (2,f(2)) 3. Find the equation of the secant line above. 4. Graph f and the secant line on GC

  7. Instantaneous Rates of Change • A billiard ball is dropped from a height of 100 ft. The function f(t) = -16t2+100 represents the height of the function in feet after t seconds. • Find the average rate of change of the function for each time interval. [1,2], [1,1.5], [1,1.1], [1, 1.01], [1,1.001] • What does the average rate of change represent? • The instantaneous rate of change is the limit of the average rates of change as the change in time becomes smaller. What is the instantaneous rate of change?

  8. Tangent lines • A tangent line to a curve y = f(x) at a point (a, f(a)) is a line that touches (not intersects) the curve at that point. • Note: it may intersect the curve at other points but it merely touches the curve at (a, f(a))

  9. Graph of a tangent line

  10. The tangent line problem • “How do we find the equation of a tangent line to a curve?” • There are many instances when this question would arise. For example in optics the tangent line is used to determine at what angle a light ray strikes a lens, and in the study of motion it is used to determine the direction a moving body is traveling. • To write the equation of a line we need a point and the slope. We have a point on the tangent line (a, f(a)) but how do we find the slope?

  11. The answer is to use secants lines and the concept of limits y = f(x) Secant Line Q(a+h, f(a+h)) Tangent Line P(a, f(a)) a a+h h

  12. As Q gets closer and closer to P (i.e. as h gets smaller and smaller) the slopes of the secant lines get closer and closer to the slope of the tangent line. See an animation of this here: http://www.ima.umn.edu/~arnold/calculus/secants/secants2/secants-j.html • In other words, the slope of the tangent line is the limit of the slopes of the secant lines as h approaches zero.

  13. Example: Analytical • Find the slope of the tangent line to the curve f(x) = x2 at (a, f(a)). • Find the slope of the tangent line at x = 2 • Find the equation of the tangent line at x = 2

  14. Example • Given the curve y = 1/x, find the point where the slope of the tangent line is –1/4. • Find the equation of the tangent line at this point.

  15. Example • A normal line to a curve y = f(x) at x = a is a line perpendicular to the tangent line at x = a and passing through the point (a, f(a)). Find the equation of the normal line to the curve y = 4 – x2 at x = 1.

  16. Example: numerical • Find the slope of the secant line PQ for the given values of x. Use the table to estimate the slope of the tangent line at x = 2 for each function. • y = x3 – x • y = • y = cos x2

  17. Position, velocity problems • If y = s(t) is a position function (i.e. it gives the position of a moving object after time) then the average rate of change of s is • Velocity is speed with direction. If the object is moving vertically – up is positive direction and down is negative direction.

  18. Instantaneous velocity • The instantaneous velocity at time t = a is or the limit of the average velocity as the time interval [a, a+h] becomes smaller and smaller.

  19. Example • The height of an object dropped from 64 ft is s(t) = 64-16t2. Find the average velocity for each time interval. Use the results to estimate the instantaneous velocity at t = 2. [1,2],[1.5,2], [1.9, 2], [1.99, 2], [1.999, 2]

  20. Example • The equation for free fall on Mars is s(t) = 1.8t2 + h0 where h0 is the height of the fall. • Find the speed of a ball dropped from 200m at time t = 1s.

  21. Position function • In general the position function of a falling object (on Earth) is where g is –16ft/s or –4.9m/s v0 is the initial velocity s0 is the initial height

  22. Example • A diver jumps from a platform 32 ft above the water at a velocity of 16 ft/s. a. When does he hit the water? b. What is the velocity at impact?

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