21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 4x²+1; a= 2,4

Verified step by step guidance

Step 1: Identify the function f(x) = 4x^2 + 1 and the values of a for which we need to evaluate the derivative, which are a = 2 and a = 4.

Step 2: Find the derivative of the function f(x) with respect to x. Use the power rule, which states that the derivative of x^n is n*x^(n-1).

Step 3: Apply the power rule to f(x) = 4x^2 + 1. The derivative of 4x^2 is 8x, and the derivative of the constant 1 is 0.

Step 4: Write the derivative function f'(x) = 8x.

Step 5: Evaluate the derivative at the given values of a. Calculate f'(2) by substituting x = 2 into f'(x), and calculate f'(4) by substituting x = 4 into f'(x).

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) and can be calculated using various rules, such as the power rule, product rule, and quotient rule.

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions in the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is calculated as n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, making it easier to evaluate derivatives at specific points.

Evaluating Derivatives at Specific Points

Evaluating the derivative at a specific point involves substituting the value of that point into the derivative function. For example, if f'(x) is found, to evaluate f'(a) for a given value of a, simply replace x with a in the derivative expression. This process yields the instantaneous rate of change of the function at that particular point, providing insights into the function's behavior.

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Textbook Question

{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

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Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

x⁴+y⁴ = 2;(1,−1)

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Textbook Question

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

b. Evaluate this derivative when a=6 and b=10.