Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(f^{\prime \prime}(2)\). $$ f(x)=5 x^{3}+2 x^{2}+x $$

Step-by-Step Solution

Answer

The second derivative at x=2 is 64.

Step 1: Differentiate Once

To find the second derivative, we first need to differentiate the function once. The given function is:\[ f(x) = 5x^3 + 2x^2 + x \]We differentiate term by term:* The derivative of \(5x^3\) is \(15x^2\).* The derivative of \(2x^2\) is \(4x\).* The derivative of \(x\) is \(1\).Thus, the first derivative is:\[ f'(x) = 15x^2 + 4x + 1 \]

Step 2: Differentiate Again

Next, we differentiate the first derivative to find the second derivative. The first derivative is:\[ f'(x) = 15x^2 + 4x + 1 \]Differentiating term by term:* The derivative of \(15x^2\) is \(30x\).* The derivative of \(4x\) is \(4\).* The derivative of \(1\) is \(0\).Thus, the second derivative is:\[ f''(x) = 30x + 4 \]

Step 3: Evaluate at x = 2

We need to find \(f''(2)\). So we substitute \(x = 2\) into the second derivative:\[ f''(2) = 30(2) + 4 \]Calculate:\[ f''(2) = 60 + 4 = 64 \]

Key Concepts

Differentiation Techniques

Differentiation is a core concept in calculus. It involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. There are several techniques used in differentiation to simplify this process.

One common method is to differentiate each term of the function individually. This means you take the derivative of each part of the equation and then combine the results. For polynomial functions like the one in our exercise, this process is straightforward.

Remember the basic rules:

• The derivative of a constant multiplied by a function is the constant times the derivative of that function.
• The power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\).

By applying these techniques, we can simplify our work and handle complex equations efficiently. Differentiation allows us to understand how a function behaves and changes over time, which is essential in many areas of math and science.

Polynomial Functions

Polynomial functions, such as the one in this exercise, are very important in calculus. They are functions that are made up of terms consisting of variables raised to non-negative integer powers, and coefficients. The general form of a polynomial function is \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\).

In our case, we have the function \(f(x) = 5 x^3 + 2 x^2 + x\). Each term can be seen as a building block. The behavior of polynomial functions is dictated by these individual terms.

• The highest power of \(x\) determines the degree of the polynomial, which also dictates its general shape and number of roots.
• Each term contributes to the overall growth and curvature of the function, influencing its slope and intercepts.

Polynomials are smooth and continuous, making them useful in modeling real-world situations that require predictions or understanding trends.

Second Derivative

The second derivative of a function is simply the derivative of its first derivative. While the first derivative gives us the rate of change of a function, the second derivative tells us about the acceleration or the concavity of the function.

Just like the first derivative, calculating the second derivative involves differentiating term by term. From our example, we found the first derivative to be \(f'(x) = 15x^2 + 4x + 1\). By differentiating this resultant function, we arrived at the second derivative: \(f''(x) = 30x + 4\).

• Positive second derivatives indicate that the function is concave up, similar to a smile.
• Negative second derivatives imply the function is concave down, like a frown.
• Where the second derivative is zero, it might indicate a point of inflection, where the concavity changes.

The second derivative is particularly useful in physics for understanding the dynamics of moving objects, as it directly relates to acceleration, providing deeper insights into the motion.