tangent

Examples of tangent in the following topics:

• Tangent Planes and Linear Approximations

  • The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • When the curve is given by $y = f(x)$ the slope of the tangent is $\frac{dy}{dx}$, so by the point–slope formula the equation of the tangent line at $(x_0, y_0)$ is:
  • The tangent plane to a surface at a given point $p$ is defined in an analogous way to the tangent line in the case of curves.
  • Note the similarity of the equations for tangent line and tangent plane.
  • Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point.

• Tangent Vectors and Normal Vectors

  • A vector is normal to another vector if the intersection of the two form a 90-degree angle at the tangent point.
  • In order for a vector to be normal to an object or vector, it must be perpendicular with the directional vector of the tangent point.
  • Tangent vectors are almost exactly like normal vectors, except they are tangent instead of normal to the other vector or object.
  • A plane can be determined as normal to the object if the directional vector of the plane makes a right angle with the object at its tangent point.
  • This plane is normal to the point on the sphere to which it is tangent.

• The Derivative and Tangent Line Problem

  • The tangent line $t$ (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • If $k$ is known, the equation of the tangent line can be found in the point-slope form:
  • Using derivatives, the equation of the tangent line can be stated as follows:
  • The line shows the tangent to the curve at the point represented by the dot.
  • Define a derivative as the slope of the tangent line to a point on a curve

• Trigonometric Functions

  • The most familiar trigonometric functions are the sine, cosine, and tangent .
  • In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (rise) of the triangle, the cosine gives the length of the x-component (run), and the tangent function gives the slope (y-component divided by the x-component) .
  • The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch).
  • The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle.

• Tangent and Velocity Problems

  • Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
  • Recognize that the slope of a tangent line to a curve gives the instantaneous velocity at that point in time

• Newton's Method

  • Geometrically, (x1, 0) is the intersection with the x-axis of a line tangent to f at (x0, f (x0)).The process is repeated as xn+1 = xn - f(xn / f'(xn) until a sufficiently accurate value is reached.
  • The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the $x$-intercept of this tangent line (which is easily done with elementary algebra).
  • The function $f$ is shown in blue and the tangent line in red.

• Linear Approximation

  • Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of $f$ at $(a, f(a))$.
  • For this reason, this process is also called the tangent line approximation.
  • Since the line tangent to the graph is given by the derivative, differentiation is useful for finding the linear approximation.

• Derivatives of Trigonometric Functions

  • The most familiar trigonometric functions are the sine, cosine, and tangent.
  • In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the $x$-axis, the sine of the angle gives the length of the $y$-component (rise) of the triangle, the cosine gives the length of the $x$-component (run), and the tangent function gives the slope ($y$-component divided by the $x$-component).
  • In this image, one can see that where the line tangent to one curve has zero slope (the derivative of that curve is zero), the value of the other function is zero.

• Derivatives and Rates of Change

  • Historically, the primary motivation for the study of differentiation was the tangent line problem, which is the task of, for a given curve, finding the slope of the straight line that is tangent to that curve at a given point.
  • The word tangent comes from the Latin word tangens, which means touching.
  • Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope.

• Direction Fields and Euler's Method

  • Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.The idea is that while the curve is initially unknown, its starting point, which we denote by $A_0$, is known (see ).
  • Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.
  • Take a small step along that tangent line up to a point, $A_1$.