posted by .

What is the definition of the slope of a curve at a point x=a?

Formal definition of a derivative?
or slope between two points?

  • math -

    the slope at a point a is the slope of the line tangent to the curve at x=a.

    To get that, pick a point number b, near a, and think of the slope between two points on the curve at f(a) and f(b).

    As b gets closer and closer to a, the line between the two points gets closer and closer to the tangent line.

    Finally, when b=a, the two points are the same point, and the line is tangent to the curve.

    That's all the formal definition of the derivative is. It's the limit of the slope of a line joining two points on the curve, as the two points get closer and closer to each other.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    The line that is normal to the curve x^2=2xy-3y^2=0 at(1,1) intersects the curve at what other point?
  2. Calculus

    show that the curves r=asin(è) and r=acos(è) intersect at right angles. can you show that the derivatives for each are the negative reciprocal of each other?
  3. Math

    How can I determine a value of gradient from a graph if the line is curved. (not a straight line) This is where Calculus comes in If a graph is a curve, then the gradient (slope) of the curve is the slope of the tangent that you draw …
  4. 12th Grade Calculus

    1. a.) Find an equation for the line perpendicular to the tangent curve y=x^3 - 9x + 5 at the point (3,5) [* for a. the answer that I obtained was y-5 = -1/18 (x-3) ] b.) What is the smallest slope on the curve?
  5. crt

    To introduce an unusual or unfamiliar word, to coin new words, or to introduce a new meaning to a familiar word." A) Stipulative definition B) Precising definition C) Explanatory definition D) Persuasive definition E) Definition by …
  6. calculus

    1. Given the curve a. Find an expression for the slope of the curve at any point (x, y) on the curve. b. Write an equation for the line tangent to the curve at the point (2, 1) c. Find the coordinates of all other points on this curve …
  7. calculus

    f(x)=1/x^2 use formal definition of derivative to find slope at x=2
  8. calculus

    1. Which of the following expressions is the definition of the derivative of f(x) = cos(x) at the point (2, cos(2))?
  9. Calc AB

    Suppose that f(x) is an invertible function (that is, has an inverse function), and that the slope of the tangent line to the curve y = f(x) at the point (2, –4) is –0.2. Then: (Points : 1) A) The slope of the tangent line to the …
  10. Math

    Find 2 tangent line equations to the curve y=x^3+x at the points where the slope of the curve is 4. What is the smallest possible slope of the curve?

More Similar Questions