sketching of derivative curve from function curve. It is in AP ... - Free Printable

To solve the problem of sketching the graph of the derivative of each given function, we need to analyze the behavior of the original functions and determine how their slopes change. Here's a step-by-step explanation for each graph:

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Graph 1:

- Original Function: A sinusoidal-like curve with multiple peaks and troughs.
- Analysis:
- The function has alternating positive and negative slopes.
- At the peaks and troughs, the slope is zero (horizontal tangent).
- Between peaks and troughs, the slope changes from positive to negative or vice versa.
- Derivative Graph:
- The derivative will be a wave-like curve that crosses the x-axis at the peaks and troughs of the original function.
- The derivative will be positive where the original function is increasing and negative where it is decreasing.
- The derivative will have local maxima and minima where the original function has inflection points.

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Graph 2:

- Original Function: A V-shaped graph (absolute value-like function).
- Analysis:
- The function has a sharp turn at \( x = 0 \).
- For \( x < 0 \), the slope is constant and negative.
- For \( x > 0 \), the slope is constant and positive.
- At \( x = 0 \), the slope is undefined.
- Derivative Graph:
- The derivative will be a step function.
- For \( x < 0 \), the derivative is a constant negative value.
- For \( x > 0 \), the derivative is a constant positive value.
- At \( x = 0 \), the derivative has a jump discontinuity.

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Graph 3:

- Original Function: A horizontal line.
- Analysis:
- The function has a constant value and no variation.
- The slope of the function is zero everywhere.
- Derivative Graph:
- The derivative will be a horizontal line at \( y = 0 \).

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Graph 4:

- Original Function: A curve that starts steeply, levels off, and then flattens out.
- Analysis:
- The function has a steep positive slope initially.
- As \( x \) increases, the slope decreases and approaches zero.
- The function appears to have a horizontal asymptote as \( x \to \infty \).
- Derivative Graph:
- The derivative will start high and positive, then decrease gradually.
- The derivative will approach zero as \( x \to \infty \).
- The derivative will be positive throughout since the original function is always increasing.

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Graph 5:

- Original Function: A straight line with a positive slope.
- Analysis:
- The function has a constant positive slope.
- Derivative Graph:
- The derivative will be a horizontal line at a positive constant value.

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Graph 6:

- Original Function: A complex curve with multiple peaks and troughs.
- Analysis:
- The function has varying slopes, including regions where the slope is positive, negative, and zero.
- There are multiple local maxima and minima.
- The function appears to oscillate rapidly in some regions.
- Derivative Graph:
- The derivative will cross the x-axis at the local maxima and minima of the original function.
- The derivative will be positive where the original function is increasing and negative where it is decreasing.
- The derivative will have rapid fluctuations corresponding to the oscillations in the original function.

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Final Answer:

The graphs of the derivatives can be sketched based on the above analyses. Here is a summary of the key features for each derivative graph:

1. Graph 1 Derivative: Wave-like curve crossing the x-axis at peaks and troughs.
2. Graph 2 Derivative: Step function with a jump discontinuity at \( x = 0 \).
3. Graph 3 Derivative: Horizontal line at \( y = 0 \).
4. Graph 4 Derivative: Decreasing curve approaching zero as \( x \to \infty \).
5. Graph 5 Derivative: Horizontal line at a positive constant value.
6. Graph 6 Derivative: Complex curve crossing the x-axis at local maxima and minima.

\[
\boxed{\text{See detailed explanations above for each graph.}}
\]