4. Applications of Derivatives

{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

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

53. lim (x → 1⁺) x^(1/(1 - x))

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.

Mean Value Theorem for quadratic functions Consider the quadratic function f(x) = Ax² + Bx + C, where A, B, and C are real numbers with A ≠ 0. Show that when the Mean Value Theorem is applied to f on the interval [a,b], the number guaranteed by the theorem is the midpoint of the interval.

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.