Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 109

Chapter 3, Problem 109

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

Verified step by step guidance

Start by identifying the function ƒ(x) = √(1 + x) + sin x - 0.5. We need to find its linearization at x = 0.

Recall that the linearization of a function at a point x = a is given by L(x) = ƒ(a) + ƒ'(a)(x - a). Here, a = 0.

Calculate ƒ(0) by substituting x = 0 into the function: ƒ(0) = √(1 + 0) + sin(0) - 0.5.

Find the derivative ƒ'(x). Use the derivative rules: the derivative of √(1 + x) is 1/(2√(1 + x)), and the derivative of sin x is cos x.

Evaluate ƒ'(0) by substituting x = 0 into the derivative: ƒ'(0) = 1/(2√(1 + 0)) + cos(0).

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Key Concepts

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

Linearization

Linearization is the process of approximating a function near a specific point using its tangent line. This involves calculating the function's value and its derivative at that point. The linear approximation can be expressed as L(x) = f(a) + f'(a)(x - a), where 'a' is the point of tangency.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of linearization, the derivative at the point of interest provides the slope of the tangent line.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For linearization, it is essential to evaluate the function and its derivative at the point of interest, which in this case is x = 0. This step is crucial for constructing the linear approximation.

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Related Practice

Textbook Question

Find the linearizations of

a. tan x at x = -π/4

Graph the curves and linearizations together.

Textbook Question

The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of

a. the radius.

b. the surface area.

c. the volume.

Textbook Question

How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

Textbook Question

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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Textbook Question

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

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Textbook Question

Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.

a. What is the relation between the variables h and r in the figure?

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