4. Applications of Derivatives

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?

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?

The diameter of a sphere is measured as 100 ± 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

Finding Functions from Derivatives

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

g'(x) = 1 / x² + 2x, P(−1, 1)

Finding Functions from Derivatives

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

r'(t) = sec t tan t − 1, P(0, 0)

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.

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

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.

Quadratic approximations

[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.