Question

Group Activity In Exercises \(27-30\) , use NDER to graph the derivative of the function. If possible, identify the derivative function by looking at the graph. $$y=-\cos x$$

Short answer

The derivative of the function \(y=-\cos x\) is \(y'=\sin x\), and the graph of the derivative function is a sinusoidal waveform, oscillating between -1 and 1.

Step-by-step solution

Calculation of the Derivative

The derivative of the function \(y=-\cos x\) is computed using the basic differentiation rules. The derivative of \(-\cos x\) is \(\sin x\). Hence, the derivative of \(y=-\cos x\) is \(y'=\sin x\).

Plotting the Graph

Now, plot the graph of the derivative function, \(y'=\sin x\). The sine function generates a wave-like graph, oscillating between -1 and 1. Since the derivative is sine, the graph will look like a sinewave.

Identifying the Derivative from the Graph

By looking at the graph of the derivative, one can identify the function as a sine function. The sinusoidal nature of the graph, oscillating between -1 and 1, confirms this.

Key concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned with the study of how things change. It's all about finding a function's derivative, which measures how the function's output value changes as its input value changes. Imagine you're driving a car, and your speedometer gives you the derivative of your position — it tells you how quickly you're moving at that very moment.

When we differentiate functions, we apply rules of differentiation to find the function that describes this rate of change. This is known as the derivative of the function. In the context of graphing, the slope of the tangent line to a curve at any given point is numerically equal to the derivative at that point. By graphing derivatives, we can visually analyze how functions increase, decrease, and where they have extrema — maximums and minimums.

Sine Function

The sine function is a fundamental trigonometric function, which is very important in a variety of scientific fields, including physics, engineering, and mathematics. It describes a smooth, periodic oscillation that repeats every \(2\pi\) radians or 360 degrees. This function takes an angle as input and outputs the y-coordinate of the corresponding point on the unit circle.

Graphically, the sine function produces a wave-like pattern, known as a sinusoid, which peaks at 1 and troughs at -1. It's symmetric with respect to origin, which means it is an odd function. This regularity and symmetry make the sine function predictable and hence, very useful in modeling periodic phenomena such as waves and vibrations.

Derivative of Cosine

The cosine function, another essential trigonometric function, relates to the sine function with a phase shift. When we take the derivative of the cosine function, we use the differentiation rules from calculus to find the rate of change. The amazing fact here is that the derivative of the cosine function, \(\cos x\), is \(\sin x\) with a negative sign. So, \(d/dx (\cos x) = -\sin x\).

It's like having a look at a mirror reflection of the sine wave when you graph it. Whenever we come across expressing the derivative of a negative cosine function, then we simply get a positive sine function (\(\sin x\)) as the derivative. It's a beautiful dance of trigonometric functions interrelated through calculus!

Plotting Graphs

Plotting graphs is a visual way of representing functions that are essential for understanding the behavior of these functions. Graphs translate complicated algebraic expressions into a form that we can see and make sense of patterns, trends, and relationships. With graphing, we move from abstract equations to concrete visual representations.

When we work with derivatives graphically, we're looking for the shape of the graph that represents how quickly the original function’s output is increasing or decreasing at each point. For instance, a sine wave, which we get from the derivative of a negative cosine function, means that our original function has maximum and minimum points where the sine wave crosses the x-axis. Plotting this graph allows us not only to see this property but also to predict function's behavior without solving complex equations each time.