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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.2.20b
Capitolo 3, Problema 3.2.20b

Use the graph of g in the figure to do the following. <IMAGE>
b. Find the values of x in (-2,2) at which g is not differentiable.

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Step 1: Understand the concept of differentiability. A function is differentiable at a point if it has a defined derivative at that point, which means the function must be continuous and smooth (no sharp corners or cusps) at that point.
Step 2: Examine the graph of the function g(x) over the interval (-2, 2). Look for points where the graph has sharp corners, cusps, or vertical tangents, as these are common places where a function is not differentiable.
Step 3: Identify any points of discontinuity within the interval (-2, 2). A function is not differentiable at any point where it is not continuous.
Step 4: Check for any vertical tangents within the interval. A vertical tangent occurs when the slope of the tangent line is undefined, which means the derivative does not exist at that point.
Step 5: List all the x-values within the interval (-2, 2) where the function g(x) is not differentiable based on the observations from the graph.

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Differentiability

A function is differentiable at a point if it has a defined derivative there, meaning the function must be continuous and have a well-defined tangent line at that point. If a function has a sharp corner, cusp, or vertical tangent, it is not differentiable at that point, even if it is continuous.
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Finding Differentials

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For differentiability, continuity is a necessary condition; however, continuity alone does not guarantee differentiability.
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Intro to Continuity

Critical Points

Critical points are values of x where the derivative of a function is either zero or undefined. These points are essential for analyzing the behavior of the function, including identifying where it may not be differentiable, such as at corners or discontinuities.
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Critical Points
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Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

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60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

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31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.

s(t)= −16t²+100t

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21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

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Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.

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Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = 1/3x-1; a= 2

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