Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.2.5

Chapter 4, Problem 4.2.5

5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

Verified step by step guidance

Step 1: Understand the Mean Value Theorem (MVT), which states that for a function ƒ that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one number c in (a, b) such that (ƒ(b) - ƒ(a)) / (b-a) = ƒ'(c).

Step 2: Identify the function ƒ(x) = x² / 4 + 1 and the interval [a, b] = [-2, 4]. Calculate ƒ(a) and ƒ(b) by substituting x = -2 and x = 4 into the function.

Step 3: Calculate the slope of the secant line using the formula (ƒ(b) - ƒ(a)) / (b-a). This represents the average rate of change of the function over the interval [a, b].

Step 4: Find the derivative of the function, ƒ'(x). For ƒ(x) = x² / 4 + 1, use the power rule to differentiate, resulting in ƒ'(x) = x / 2.

Step 5: Set the derivative equal to the slope of the secant line: x / 2 = (ƒ(b) - ƒ(a)) / (b-a). Solve this equation for x to find the value(s) of c that satisfy the Mean Value Theorem.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Secant Line

A secant line is a straight line that intersects a curve at two or more points. In calculus, it is often used to approximate the slope of the curve between those points. The slope of the secant line between points (a, f(a)) and (b, f(b)) is given by the formula (f(b) - f(a)) / (b - a), which represents the average rate of change of the function over the interval [a, b].

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the derivative at that point equals the average rate of change over the interval. This theorem is fundamental in connecting the concepts of secant lines and instantaneous rates of change, as it provides a formal justification for finding a point where the slope of the tangent line (f'(c)) matches the slope of the secant line.

Derivative

The derivative of a function at a point measures the instantaneous rate of change of the function with respect to its variable at that point. It is defined as the limit of the average rate of change as the interval approaches zero. In the context of the question, f'(c) represents the slope of the tangent line to the curve at the point c, which is crucial for verifying the relationship established by the Mean Value Theorem.

Recommended video:

05:44

Derivatives

Related Practice

Textbook Question

Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = x³ - 13x² - 9x

159

views

Textbook Question

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

lim_x→π (cos x +1 ) / (x - π )²

202

views

Textbook Question

Crankshaft A crank of radius r rotates with an angular frequency w It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function <IMAGE>

a (Θ) = w²r (cos Θ + (r cos2Θ) / L) .

For fixed w , L, and r find the values of Θ, with 0 ≤ Θ ≤ 2π , for which the acceleration of the piston is a maximum and minimum.

221

views

Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = (x - 1)²

269

views

Textbook Question

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.

ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .

219

views