Question

Identify a function \(f\) that has the given characteristics. Then sketch the function. $$ f(0)=2 ; f^{\prime}(x)=-3,-\infty

Short answer

The function that satisfies the given characteristics is \(f(x) = -3x+2\).

Step-by-step solution

Interpret the derivative

The derivative of a function measures the rate of change at each point on the function. As per the given characteristic \(f'(x) = -3\), for all real numbers (\(-\infty < x < \infty\)). This implies that the function \(f\) is a linear function whose slope is -3.

Identify the function from derivative

Since we have established that the function \(f\) must be a linear function, the general form of a linear function is \(f(x) = mx+b\), where 'm' is the slope and 'b' is the y-intercept. For our function, the slope 'm' is -3 (the derivative). So, the function has now the form \(f(x) = -3x+b\).

Use the given point to find 'b'

The given characteristic \(f(0)=2\) means the function crosses the y-axis at the point (0,2). This gives the value of 'b' in the function \(f(x) = -3x+b\). Substituting \(x = 0\) and \(f(0) = 2\) into the equation we get 2 = -3*0+b or b=2. Therefore, the function is \(f(x) = -3x+2\).

Sketch the function

To sketch the function \(f(x) = -3x+2\), draw a straight line with a slope of -3 (downwards as we move from the left to right) and intersecting the y-axis at the point (0,2). The line represents the function \(f\).

Key concepts

Functions and Derivatives

Understanding the relationship between functions and their derivatives is a key concept in calculus. When a function, represented as f(x), describes a certain quantity in relation to another (often time or distance), its derivative f'(x) represents how this quantity changes with a small increase in x. Essentially, the derivative gives the rate of change, or the slope, at any point along the function.

For linear functions, the derivative is constant and equals the slope of the line. In the exercise, the derivative of the function f is given as f'(x) = -3. This indicates a constant rate of change; for every unit that x increases, f(x) decreases by 3. Since the slope is negative, we know the function is decreasing. The constant nature of the derivative in this situation tells us that f is a linear function.

Rate of Change

The rate of change is a fundamental concept that measures how a quantity changes in response to changes in another quantity. In calculus, the rate of change is precisely captured by the derivative. For the given exercise, since f'(x) = -3, we know that the function decreases by 3 units for every increase of 1 unit in x.

In everyday terms, if you were driving and your speedometer read a constant speed, that speed would be analogous to the derivative of your position over time - a linear relationship. If we graphed your distance over time, the slope of the line would reflect your speed (rate of change). Similarly, in our exercise, the rate of change is constant and negative, which will be evident in the graph of the function.

Linear Function Graphing

Graphing a linear function is a straightforward process once you've identified the slope and y-intercept. The function from our exercise is f(x) = -3x + 2. To graph this, start at the y-intercept (0,2), which is where the function intersects the y-axis, indicating the starting value of f(x) when x is zero.

To continue the graph, use the slope of -3. From each point on the line, you can move three units down (because the slope is negative) and one unit to the right (the rise over run) to find the next point. Drawing a straight line through these points will complete the graph. The downward slope reflects the negative rate of change, confirming that the function decreases as x increases.