a) What is a derivative? What information can it provide us? b) What is the limi
ID: 2844144 • Letter: A
Question
a) What is a derivative? What information can it provide us?
b) What is the limit definition of the derivative? From what basic formula relating to lines is it developed? How is that related to part a?
c) Use the limit definition of the derivative to find the derivative of the following function.
f(x) = 2x^3
d) Given that the generic form of an objects path through time in relationship to earth is
s(t)=-16t^2+Vot+so Do the following:
a) Find the model for the Velocity of the object through time where the initial velocity is 50ft/sec and the initial height is 600ft.
b) Find the average velocity of the object between t = 1 and t = 4 seconds
c) Find the instantaneous velocity of the object at t = 2 seconds
e) Remembering, that the derivative is equal to the slope of the tangent line to the curve at any point on the curve, do the following:
a) Plot f(x) = x^2 and f '(x) = 2x in your graphing calculator
b) How does f
Explanation / Answer
a) What is a derivative? What information can it provide us?
b) What is the limit definition of the derivative? From what basic formula relating to lines is it developed? How is that related to part a?
A derivative of a function tells us how fast the output variable, y, is changing compared to the input variable, x. For example, if y is increasing 2 times as fast as x (like with the line y = 2x + 7), then we say that the derivative of y to the respect to x equals 2, and we write dy/dx = 2, which is the same as dy/dx = 2/1. That means that we can say that the rate of change of y compared to x is 2:1, or that the line has a slope of 2/1. We can think of a derivative dy/dx as basically rise/run. So, the derivative is basically just a rate or a slope. Thus, to solve a problem, all we have to do is answer the question as if it asks us to determine a rate or a slope instead of a derivative. Any line of the form y = mx + b has a slope equal to m.
Two problems which greatly influenced the development of the differential calculus are:
1. finding the equation of the tangent line to a given curve at a given point on the curve, and
2. finding the instantaneous velocity of a particle moving along a straight line at a varying speed (a derivative is always a rate, and a rate is always a derivative, assuming we are talking about instantaneous rates).
So, one way to think about a derivative like dp/dt is that it tells us how much the position, p, changes when the time, t, increases by a specific time. For example, a driver starts at a time=0, and go 50 miles/hour in his car. The rate of 50 miles/hour means that his position changes 50 miles each time the number of hours of his trip goes up by 1.
Using the information from the problem above, we can write a function that gives us the driver's position as a function of time.
p(t) = 50t or p = 50t, where p is in miles and t is in hours.
p = 50t is a line, of course, in the form of y = mx + b (where b = 0). So,the slope is 50 and the derivative also is 50. And again we see that a derivative is a slope and a rate.
We noticed in a plane geometry that a straight line intersects a circle in two points, or is tangent to the circle, or fails to intersect the circle at all. This might tempt us to define a tangent to a circle as a line that intersects the circle in one and only one point.
But such a definition would not do for most other curves. For example, the tangent line to the graph of y = x^3 at the point (1, 1) intersects the curve again at the point (-2, -8). This indicate that a different approach is needed.
Since we can write the equation of a line through a given point if we know the slope of the line, our task is to formulate a definition of the slope of the tangent to a curve which will apply to all curves as well.
The difference quotient is a magnificent tool that gives us the slope of a curve at a single point. For example, if we have a parabola and pick a point on it, let say the point (2, 4). We can't get the slope of the parabola at (2, 4) with algebra slope formula, m = (y2 - y1)/(x2 - x1), because no matter what other point on parabola we use with (2, 4) in the formula, we will get a slope that is steeper or less steep than the precise slope at (2, 4).
But, if our second point on the parabola is extremely close to (2, 4), for example the point (2.001, 5.00299...), our line would be almost exactly as steep as the tangent line. The difference quotient gives the precise slope of the tangent line by sliding the second point closer and closer to (2, 4) until its distance from (2, 4) is infinitely small.
The definition of the derivative based on the difference quotient is:
f'(x) = lim h -->0 = [f(x + h) - f(x)]/h
[In the example above h = 2.001 - 2 = 0.001, that is "delta x" and its symbol is ?x. By the symbol ?x we mean by how much the x-coordinate will change (as we move from A(2, 5) to B(2.001, 5.00299...). And by the symbol ?y ("delta y") we mean by how much the y-coordinate will change.
Given any number x for which the limit exists, we assign to x the number f'(x). So, we can regard f' as a new function, called the derivative of f and defined by the above formula. We know that the value of f' at x, f'(x), can be interpreted geometrically as the slope of the tangent line to the graph of f at the point (x, f(x)).
The function f' is called the derivative of f because it has been derived from by limiting operation in the equation (formula) above. The domain of f' is the set {x| f'(x) exists} and may be smaller than the domain of f.
Definition: If f is a function and P(c, f(c)) is a point on the graph of y = f(x) the slope of the tangent to the graph at P(c, f(c)) is
lim ?x -->0 ?y/?x = lim ?x -->0 [f(c + ?x) - f(c)]/?x
provided that this limit exists.
The above limit is called the value of the derivative of the function f at c.
If f is a position function which gives the coordinate s = f(t) at time t of a particle moving along a coordinate line (Fig. below),
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