Problem 23 - 5 Steps to A 5: AP Calculus AB 2021

Question

Find \(\frac{d^{2} y}{d x^{2}}\) if \(y=\cos (2 x)+3 x^{2}-1\).

Step-by-Step Solution

Answer

Answer: \(\frac{d^2 y}{dx^2} = -4\cos(2x) + 6\)

Step 1: Find the first derivative, \(\frac{dy}{dx}\)

To find the first derivative, differentiate each term of the function with respect to x: $$ \frac{d}{dx}\cos(2x) = -2\sin(2x) $$ $$ \frac{d}{dx}(3x^2) = 6x $$ $$ \frac{d}{dx}(-1) = 0 $$ Combining these, we find the first derivative: $$ \frac{dy}{dx} = -2\sin(2x) + 6x $$

Step 2: Find the second derivative, \(\frac{d^2 y}{dx^2}\)

Now, differentiate the first derivative with respect to x to find the second derivative: $$ \frac{d}{dx}(-2\sin(2x)) = -4\cos(2x) $$ $$ \frac{d}{dx}(6x) = 6 $$ Combining these, we find the second derivative: $$ \frac{d^2 y}{dx^2} = -4\cos(2x) + 6 $$ The second derivative of \(y = \cos(2x) + 3x^2 - 1\) with respect to x is \(\frac{d^2 y}{dx^2} = -4\cos(2x) + 6\).

Key Concepts

Understanding Calculus Differentiation

Calculus is a branch of mathematics that helps us understand changes between values that are related by a function. Differentiation, a key concept in calculus, is used to calculate the rate of change of a function with respect to one of its variables. In simpler terms, differentiation helps find how a function's output changes when we slightly change its input.
For example, if we have a function like \( y = \cos(2x) + 3x^2 - 1 \), we use differentiation to find how the value of \( y \) changes as \( x \) changes. This involves finding the first derivative \( \frac{dy}{dx} \), which represents the slope of the tangent line at any point on the function.

• It helps us determine the increasing or decreasing nature of the function.
• Allows us to find optimization points, like maximum or minimum values.

Understanding calculus differentiation is crucial for solving real-world problems where change is involved, from physics to economics.

Exploring Trigonometric Differentiation

Trigonometric differentiation is a special case of differentiation that applies to functions including trigonometric functions like sine, cosine, and tangent. These functions have specific rules for how they change in response to changing inputs.
In the given exercise, we dealt with \( \cos(2x) \) as part of the function. The differentiation of cosine and sine follows these simple rules:

• The derivative of \( \sin(ax) \) is \( a\cos(ax) \).
• The derivative of \( \cos(ax) \) is \(-a\sin(ax) \).

By applying these, we get the first derivative as \( -2\sin(2x) \). Understanding these rules makes it straightforward to differentiate trigonometric functions, assisting in reaching correct solutions efficiently.

The Role of Higher Order Derivatives

Higher order derivatives are derivatives of derivatives. The initial derivative gives the rate of change of the original function, and a second derivative tells us how this rate itself is changing. This concept is fundamental in analyzing the curvature of graphs.

• The second derivative \( \frac{d^2y}{dx^2} \) can tell us about the concavity of the function. If it is positive, the function is concave up, like a smiling face. Conversely, a negative second derivative implies concave down.
• They are also useful in physics for understanding acceleration when we start from a position-time graph.

In our example \( \frac{d^2y}{dx^2} = -4\cos(2x) + 6 \), helps us identify how the function’s curvature behaves as \( x \) changes, offering deeper insights into the function's behavior across its domain.