4. Applications of Derivatives

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).

Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.

Explain the Mean Value Theorem with a sketch.

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = ln x - x² + 3x - 1

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_z→∞ (1 + 10/z²)ᶻ²

If $f\left(x\right)=x^3-8x+6$, find the differential $dy$ when x = 2 and $dx = 0.2$.

If $f\left(x\right)=\sqrt{x+2}$, find the differential $dy$ when $x=2$ and $dx=0.01$.

Lapse rates in the atmosphere Refer to Example 2. Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is -10.3° C and at an elevation of 3.2km , the temperature is 8.0°C . Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of 7°C/ km at some intermediate elevation? Explain.

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]

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) = sin 2x; [0, π/2]

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) = 1 - | x | ; [-1, 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) = 1 - x²⸍³ ; [-1, 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.

g(x) = x³ - x² - 5x - 3; [-1, 3]

Let ƒ(x) = x²⸍³ . Show that there is no value of c in the interval (-1, 8) for which ƒ' (c) = (ƒ(8) - ƒ (-1)) / (8 - (-1)) and explain why this does not violate the Mean Value Theorem.

Running pace Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runner’s speed at the finish line is zero.