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Taylor Polynomials and Series for Math 125
(x) is the polynomial of degree two that has the same function value at x = a, the same first derivative value at x = a, and the same second derivative value at x = a as the original function f(x). Example 1.1 Find the Taylor polynomials of degrees …

[PDF File]Section 14.5 (3/23/08) Directional derivatives and ...
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The graph of this function is shown in the sz-plane of Figure 4. The slope of its tangent line at s = 0 is the directional derivative from Example 1. The corresponding cross section of the surface z = f(x,y) is the curve over the s-axis drawn with a heavy line in …

[PDF File]The First and Second Derivatives - Dartmouth College
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For an example of ﬁnding and using the second derivative of a function, take f(x) = 3x3 ¡ 6x2 + 2x ¡ 1 as above. Then f0(x) = 9x2 ¡ 12x + 2, and f00(x) = 18x ¡ 12. So at x = 0, the second derivative of f(x) is ¡12, so we know that the graph of f(x) is concave down at x = 0. Likewise, at x = 1, the second derivative of f(x) is f00(1) = 18 ¢1¡12 = 18¡12 = 6; so the graph of f(x) is ...

[PDF File]Lesson 2.6: Diﬀerentiability
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Afunctionisdiﬀerentiable at a point if it has a derivative there. In other words: The function f is diﬀerentiable at x if lim h→0 f(x+h)−f(x) h exists. Thus, the graph of f has a non-vertical tangent line at (x,f(x)). The value of the limit and the slope of the tangent line are the derivative of f at x …

[PDF File]Rules for Finding Derivatives - Whitman College
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graph that is a horizontal line, with slope zero everywhere. ... just remember that the derivative of any constant function is zero. Exercises 3.1. Find the …

[PDF File]Derivative of arcsin(x) - MIT OpenCourseWare
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Derivative of arcsin(x) For a ﬁnal example, we quickly ﬁnd the derivative of y = sin−1 x = arcsin x. As usual, we simplify the equation by taking the sine of both sides: y = sin−1 x sin y = x We next take the derivative of both sides of the equation and solve for y …

[PDF File]Average and Instantaneous Rates of Change: OBJECTIVES The ...
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The function that gives this instantaneous rate of change of a function f is called the derivative of f. If f is a function deﬁned by then the derivative of f(x) at any value x, denoted is if this limit exists. If exists, we say that f is differentiable at c. The following procedure illustrates how to ﬁnd the derivative of a function at any ...

[PDF File]Chapter Nine: Profit Maximization
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A graph showing a profit curve that has an inverted U-shape and has a peak at the profit maximizing quantity. Profit is maximized at the quantity q* and is lower at all other quantities. The curvature of the profit function is consistent with a negative second derivative and results in q* being a quantity of maximum profit.

[PDF File]Instantaneous Rate of Change — Lecture 8. The Derivative.
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prisingly, we call this new function the derivative of f(x). Thus, The derivative of a function y = f(x) is the function deﬁned by f0(x) = lim h→0 f(x+h)−f(x) h. So the derivative f0(x) of a function y = f(x) spews out the slope of the tangent to the graph y = f(x) at each x in the domain of f where there is a tangent line.

[PDF File]Derivatives of inverse function PROBLEMS and SOLUTIONS
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Derivative of the inverse function at a point is the reciprocal of the derivative of the function at the corresponding point . Slope of the line tangent to 𝒇 at 𝒙= is the reciprocal of the slope of 𝒇 at 𝒙= . 1. Find tangent line at point (4, 2) of the graph of f -1 if f(x) = x3 + 2x – 8 2.

Derivative of a function graph