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Any function that can't be drawn as a straight line will have a different slope at consecutive point. If it has to have a different slope at every point, the function constantly increasing or decreasing with a positive or negative concavity everywhere. Function of the form y=a^x and y=log(x) fit this description perfectly. Functions of the odd roots of x would also display similar behavior.
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What is the slope of a curve line?
The slope of a curved line changes as you go along the curve and so you may have a different slope at each point. Any any particular point, the slope of the curve is the slope of the straight line which is tangent to the curve at that point. If you know differential calculus, the slope of a curved line at a point is the value of the first derivative of the equation of the curve at that point. (Actually, even if you don't know differential calculus, the slope is still the value of the function's first derivative at that point.)
How can you prove that 2 equations ...x squared plus y squared equals 25... and a linear equation ...y equals -.75x plus 6.. are tangent to each other algebraically and without a calculator?
The solutions (x,y) to the first equation form the circle centered at 0, with radius 5 (the square root of 25). Solving for y, you can see that it consists of two functions of x: y = sqrt(25-x^2), y = -sqrt(25-x^2). Now you must show that y = -0.75x + 6 is tangent to one of them. That is, show that for one of the curves, at some point x', the slope is -0.75, AND the line intersects the curve at x' (that is, the y-values are equal for the curve and line, at point x'). Compute the derivative of each curve, and find for what x' it is equal to -0.75. (For each derivative, there is an easy integer solution which will work. No calculator needed.) Then check if the line intersects the curve at the point x' you find (for each curve). (It will work for one of the curves.) So you just showed that at point x', one of the curves has slope -0.75, and the line (also with slope -0.75) intersects the curve at x'. Therefore, the line is tangent to the curve at point x'. Hope that helps.
A plane curve with each point on the curve equidistant fom fixed point at the center of figure?
A circle.You don't even need the words " ... at the center of the figure".
What does delta sign d mean?
A delta δ is used to indicate a small change in a variable. -------------------------------------------------------------- In calculus it "mutates" into a 'd' to represent differentiation which occurs when the small change is over another small change and both tend to zero. For example, when calculating the slope of curve at a point, a small chord is drawn from that point (x, y) to a nearby point (x + δx, y + δy) also on the curve; the slope of this line is δy/δx and approximates to the slope of the curve at (x, y) - the closer (x + δx, y + δy) is to (x, y), ie the smaller δx (and thus δy) is, the closer the slope of the chord is to the slope at the point itself. As δx→0, δy→0 the slope of the chord tends to 0/0 at the point itself; but each point of the curve has a slope and this "mutates" into dy/dx and becomes differentiation (of the curve at that point). eg Consider the quadratic y = ax² + bx + c The slope at point (x, y) can be found by considering the chord to (x + δx, y + δy) and letting δx→0. y + δy = a(x + δx)² + b(x + δx) + c → δy = a(x + δx)² + b(x + δx) + c - y = a(x² + 2xδx + δx²) + b(x + δx) + c - (ax² + bx + c) = ax² + 2axδx + aδx² + bx + bδx + c - ax² - bx - c = 2axδx + bδx + aδx² = δx(2ax + b + aδx) → slope chord = δy/δx = δx(2ax + b + aδx)/δx = 2ax + b + aδx (as δx ≠ 0) → The slope at (x, y) = 2ax + b + aδx as δx→0. Now let δx = 0 → slope at (x, y) = dy/dx = 2ax + b + a.0 = 2ax + b
How do you tell whether a graph shows a constant or variable rate of change?
The slope of each point on the line on the graph is the rate of change at that point. If the graph is a straight line, then its slope is constant. If the graph is a curved line, then its slope changes.
What is the slope of a curve line?
The slope of a curved line changes as you go along the curve and so you may have a different slope at each point. Any any particular point, the slope of the curve is the slope of the straight line which is tangent to the curve at that point. If you know differential calculus, the slope of a curved line at a point is the value of the first derivative of the equation of the curve at that point. (Actually, even if you don't know differential calculus, the slope is still the value of the function's first derivative at that point.)
What does the slope of the curve on a distance v time graph represent?
The slope of the curve at each point on thegraph is the speed at that point in time. (Not velocity.)
How do you find the slope of the tangent line to each curve at the given point?
By differentiating the answer and plugging in the x value along the curve, you are finding the exact slope of the curve at that point. In effect, this would be the slope of the tangent line, as a tangent line only intersects another at one point. To find the equation of a tangent line to a curve, use the point slope form (y-y1)=m(x-x1), m being the slope. Use the differential to find the slope and use the point on the curve to plug in for (x1, y1).
What is the definition of the tangency condition in relation to curves and how does it affect the behavior of the curve at the point of tangency?
The tangency condition refers to the point where a curve and a straight line touch each other without crossing. At this point, the curve and the line have the same slope. This affects the behavior of the curve at the point of tangency by creating a smooth transition between the curve and the line, without any abrupt changes in direction.
How to measure the point at which two tangents intersect each other?
To measure the point at which two tangents intersect each other, find an equation for each tangent line and compute the intersection. The tangent is the slope of a curve at a point. Knowing that slope and the coordinates of that point, you can determine the equation of the tangent line using one of the forms of a line such as point-slope, point-point, point-intercept, etc. Do the same for the other tangent. Solve the two equations as a system of two equations in two unknowns and you will have the point of intersection.
What is the equation of a line in general form with a point -3 0 and an undefined slope?
You can have infinitely many lines through one specific point, each with a different equation. If you want to have a general equation for ANY line that goes through that point, use the point-slope equation for a line, and use a variable for the slope.
What do the points on a market supply curve represent?
Each point on a market supply curve denotes basically the same thing. Each point on the curve corresponds to the supply of something, but at a specific or given price.
A six letter word for a plane curve with each point on the curve being and equal distance from a fixed point at the center of the figure?
The distance from the fixed point at the center of a circle to any point on the curve is called the radius.
How can you prove that 2 equations ...x squared plus y squared equals 25... and a linear equation ...y equals -.75x plus 6.. are tangent to each other algebraically and without a calculator?
The solutions (x,y) to the first equation form the circle centered at 0, with radius 5 (the square root of 25). Solving for y, you can see that it consists of two functions of x: y = sqrt(25-x^2), y = -sqrt(25-x^2). Now you must show that y = -0.75x + 6 is tangent to one of them. That is, show that for one of the curves, at some point x', the slope is -0.75, AND the line intersects the curve at x' (that is, the y-values are equal for the curve and line, at point x'). Compute the derivative of each curve, and find for what x' it is equal to -0.75. (For each derivative, there is an easy integer solution which will work. No calculator needed.) Then check if the line intersects the curve at the point x' you find (for each curve). (It will work for one of the curves.) So you just showed that at point x', one of the curves has slope -0.75, and the line (also with slope -0.75) intersects the curve at x'. Therefore, the line is tangent to the curve at point x'. Hope that helps.
What is the point where the consumer can get the most satisfaction and which at the same time he or she can afford?
The point of tangecy between the consumers buget constraint and highest atainable indiference curve. To find this you make the slopes equal each other. The slope of the indiference curve is the marginal rate of substitution (mux/muy) and the slope of the buget constraint is the price of the horizontal good divided by the price of the virtical good (px/py). Mux/muy=px/py
Why an indifference curve has a negative slope?
As quantity consumed of one good (X) increases, total utility (satisfaction) would increase if not offset by a decrease in the quantity consumed of the other good (Y). Satisfaction, or utility must be offset so that at each point on the curve 'indifference' is retained.
What do you call the point where a line on a graph changes slope?
Nothing particular. The graph of y = x2, for example, changes slope at each point on the graph.
