When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.


b. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain.

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

r'(θ) = 8 − csc²θ, P(π/4, 0)

Applications

Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour).

Applications

A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.

a. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c).

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.

Quadratic approximations

b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.

Quadratic approximations

d. Find the quadratic approximation to g(x) = 1/x at x = 1. Graph g and its quadratic approximation together. Comment on what you see

Quadratic approximations

[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.