BackAs a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)
Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
b. Solve the equation ƒ(𝓍) = 0 graphically with an error of magnitude at most 10⁻⁸ .
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
c. It can be shown that the exact value of the solution in part (b) is
(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³
Evaluate this exact answer and compare it with the value you found in part (b).
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = x√(1 − x²)
Derivatives in Differential Form
In Exercises 17–28, find dy.
2y³/² + xy − x = 0
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sec(x² − 1)
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sin(5√x)
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = 3 csc(1 − 2√x)
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
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f(x) = x² + 2x, x₀ = 1, dx = 0.1
The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s
b. area?
The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s
a. circumference?
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change