BackApproximating changes
Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = sin x + x - 1; x₀ = 0.5
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
ƒ(x) = ln 2x; [1,e]
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2π (x sin x + x² - 4π²) / (x - 2π)
Finding Position from Velocity or Acceleration
Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.
v = 9.8t + 5, s(0) = 10
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
eˣ and 3ˣ
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sec(x² − 1)
Tolerance
b. About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_y→2 (y²+y-6) / (√(8-y²)-y)
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = 2x cos x on [0,2]
Means
b. Show that the point guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a,b], where 0 < a < b, is the geometric mean of a and b; that is, c = √ab.
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
ƒ(x) = x (x - 1)² ; [0, 1]