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2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming
If f( t ) represents the position of an object as a function of time, then the rate of change is the velocity of the object. Average rate of change (from bc) Average rate of change of f(x) over the interval [a,b]
Find the average rate of change of f ( t ) = 2 + cos t on [0, π ] F(b)= f( π ) = 2 + cos ( π ) = 2 – 1 = 1 F(a)= f( 0) = 2 + cos (0) = 2 + 1 = 3 1. Calculate the function value (position) at each endpoint of the interval The average velocity on [0, π ] is 0.63366 2. Use the slope formula
Consider a graph of displacement (distance traveled) vs. time. Average velocity can be found by taking: The speedometer in your car does not measure average velocity, but instantaneous velocity. The velocity problem time (hours) distance (miles) A B (The velocity at one moment in time.)
The slope of a line is given by: The slope of a curve at (1,1) can be approximated by the slope of the secant line through (1,1) and (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).
The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?
If we try to apply the same formula to find The instantaneous velocity and evaluate the velocity at an instant (a,f(a)) not an interval , we will find it Which is undefined, so the best is to make Δ x as small as experimentally possible Δ x 0 The instantaneous velocity at the point (a,f(a)) =
The slope of the secant line= the average rate of change = The average velocity The slope of the curve at a point = the slope of the tangent line of the curve at this point = instantaneous velocity =
Other form for Slope of secant line of tangent line Let h = x - a Then x = a + h
Rates of Change: These definitions are true for any function. ( x does not have to represent time. ) Average rate of change = Instantaneous rate of change =
Rewrite algebraically if direct substitution produces an indeterminate form such as 0/0
Factor and reduce
Rationalize a numerator or denominator
Simplify a complex fraction
When you rewrite you are often producing another function that agrees with the original in all but one point. When this happens the limits at that point are equal.
Find the indicated limit = - 5 direct substitution fails Rewrite and cancel now use direct sub. 0 0
slope slope at The slope of the curve at the point is:
If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use. The slope of the curve at the point is: is called the difference quotient of f at a .
The slope of a curve at a point is the same as the slope of the tangent line at that point. (The normal line is perpendicular.) In the previous example, the tangent line could be found using . If you want the normal line , use the negative reciprocal of the slope. (in this case, )
Example 4: Calc a Find the slope at . Let On the TI-89: limit ((1/( a + h ) – 1/ a ) / h , h , 0) F3 Note: If it says “Find the limit” on a test, you must show your work !
Example 4: y = 1 / x b Where is the slope ? Let On the TI-89: Y= WINDOW GRAPH
Example 4: y = 1 / x We can let the calculator plot the tangent: Math A: Tangent 2 Repeat for x = -2 tangent equation b Where is the slope ? Let On the TI-89: Y= WINDOW GRAPH F5 ENTER ENTER
Review: average slope: slope at a point: average velocity: instantaneous velocity: So are these! If is the position function: These are often mixed up by Calculus students! velocity = slope