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04:57
5:10 04:57Differentiability and Continuity on an Interval
Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be
a. differentiable?
Give reasons for your answers.
Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.
a. f(x) = (1 − x)⁶
Tolerance
a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value?
In Exercises 47 and 48, find an equation for
(a) the tangent line to the curve at P
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Quadratic approximations
a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:
i. Q(a) = f(a)
ii. Q′(a) = f′(a)
iii. Q″(a) = f″(a).
Determine the coefficients b₀, b₁, and b₂.
Differentiability and Continuity on an Interval
Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be
a. differentiable?
Give reasons for your answers.
