4. Applications of Derivatives

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cosⁿ x) / x²

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).

{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) = x² - 10; x₀ = 3

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the volume V = x³ of a cube when the edge lengths change from x₀ to x₀ + dx

{Use of Tech} Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of ³√10, we let x = ³√10 and cube both sides of the equation to obtain x³ = 10, or x³ - 10 = 0. Therefore, ³√10 is a root of p(x) = x³ - 10, which we can approximate by applying Newton’s method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of x₀ and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.

r = 7¹/⁴

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 

c. Graph g for a = 2, 3, and 4.

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = x²ᐟ³, [−1, 8]

{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) = x ln (x + 1) -1 ; x₀ = 1.7

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

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

Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

i. y = x² − 4

Estimating height of a building A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?

To find the height of a lamppost (see accompanying figure), you stand a 6-ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.

9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)