![]() We will learn later what these higher order derivatives are used for. Misconception 2: A tangent line to a curve must cross 2.2-2.2 h. (y double prime) is the third derivative. Solution for Tangent Lines and the Derivative Homework Name 2.2 Date Period Problems 1-6 Find lim Find answers to questions asked by students like you. Is the first derivative of y with respect to x. Example: f(x) = | x |ĥ To be differentiable, a function must be continuous and smooth.ĭerivatives will fail to exist at: corner cusp discontinuity vertical tangent Note: The converse is false: there are functions that are continuous but not differentiable. Theorem: If f is differentiable at a, then f is continuous at a. It is differentiable on an open interval (a,b) if it is differentiable at every number in the interval. dy / dx should not be regarded as a ratio.Ģ The derivative is the slope of the original function.Ī function f is differentiable at a if f ′(a) exists. Find an equation of the tangent line at x 3. Sketch the line tangent to g (x) at x- 1.57 on the graph above. 1 -2 It is a fact that the derivative of this function is g (x) cosx-x sinx. If we replace number a by a variable x, then the derivative can be interpreted as a function of x : Alternative notations for the derivative: D and d / dx are called differentiation operators. Question: GROUP WORK 1, SECTION 2.2 Tangent Lines and the Derivative Function The following is a graph of g (x)- x cosx. ![]() 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line. 3) Plug x value into f(x) to find the y coordinate of the tangent point. ![]() In Section 2.1 we considered the derivative at a fixed number a. 2) Plug x value of the indicated point into f '(x) to find the slope at x.
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