Graph Function and Its Derivative - Calculus I with Applications ... - Free Printable

I will solve this problem by matching each function graph (a)-(d) with the graph of its derivative (i)-(iv). The key principle is that the derivative represents the slope of the original function. Where the function is increasing, the derivative is positive; where it's decreasing, the derivative is negative; and at local maxima/minima (where the slope is zero), the derivative crosses the x-axis.

Let me analyze each function one by one.

Step 1: Analyze Function (a)

Looking at graph (a), I can see:
- It starts by decreasing (negative slope)
- Then increases to a local maximum (positive slope)
- Then decreases again (negative slope)

This means the derivative should start negative, cross zero to become positive, then cross zero again to become negative. This pattern matches graph (iii).

Step 2: Analyze Function (b)

Graph (b) shows a piecewise linear function with sharp corners.
- On the left, it has a constant positive slope
- Then a constant negative slope
- Then a constant positive slope again

Since the slopes are constant in each segment, the derivative should be horizontal lines (constant values) in each segment. The derivative will have discontinuities at the corners. This matches graph (iv), which shows horizontal lines with open circles indicating discontinuities.

Step 3: Analyze Function (c)

Graph (c) is symmetric about the y-axis and has a minimum at x=0.
- For x < 0, the function is decreasing (negative slope)
- At x = 0, the slope is zero (minimum point)
- For x > 0, the function is increasing (positive slope)

The derivative should be negative for x < 0, zero at x = 0, and positive for x > 0. Looking at the options, graph (i) matches this description: it's negative on the left, crosses zero at the origin, and is positive on the right.

Step 4: Analyze Function (d)

Graph (d) is also symmetric about the y-axis but has a maximum at x=0.
- For x < 0, the function is increasing (positive slope)
- At x = 0, the slope is zero (maximum point)
- For x > 0, the function is decreasing (negative slope)

The derivative should be positive for x < 0, zero at x = 0, and negative for x > 0. This matches graph (ii), which is positive on the left, peaks at zero, and is negative on the right.

Final Answer:

- (a) matches with (iii)
- (b) matches with (iv)
- (c) matches with (i)
- (d) matches with (ii)

The reasoning is based on the relationship between a function and its derivative: the derivative represents the slope of the tangent line to the function at each point. By analyzing where each function is increasing or decreasing, we can determine the sign and behavior of its derivative.