Grade 11 · Rational functions · 57 of 600
Rational functions · Numerator a=3.5; Pole location b=1
Evaluate at x=3.5, exactly 2.5 units to the right of the pole. Find the value, derivative, and excluded input. Givens: Numerator a=3.5; Pole location b=1.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(x)=3.5/(x−(1)), with x≠1. Evaluate at x=3.5, exactly 2.5 units to the right of the pole. Find the value, derivative, and excluded input.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Use the distance x−b as the denominator. The derivative is −a/(x−b)².
Work through the reasoning
Step 1
Identify the model and target
Let f(x)=3.5/(x−(1)), with x≠1. Evaluate at x=3.5, exactly 2.5 units to the right of the pole. The governing relation is f(x)=a/(x−b); f′(x)=−a/(x−b)². Use the distance x−b as the denominator. The derivative is −a/(x−b)².
Step 2
Substitute and calculate
The denominator is 3.5−(1)=2.5, so 3.5/(2.5)=1.4. Its derivative is −3.5/(2.5)²=-0.56.
Step 3
Check the mathematical meaning
The denominator is 2.5>0, so the value 1.4 is positive. The derivative -0.56 is negative. The excluded input remains 1, not the current moving input. Input x: 3.5; Reciprocal value: 1.4; Slope: -0.56; Excluded input: 1. Decimal values are rounded, so use unrounded intermediate values.
The answer
The denominator is 3.5−(1)=2.5, so 3.5/(2.5)=1.4. Its derivative is −3.5/(2.5)²=-0.56. The denominator is 2.5>0, so the value 1.4 is positive. The derivative -0.56 is negative. The excluded input remains 1, not the current moving input. Animation check: Input x: 3.5; Reciprocal value: 1.4; Slope: -0.56; Excluded input: 1. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
The mathematical idea
A shifted reciprocal has an excluded input at x=b and approaches the horizontal axis far from that input. A positive numerator makes the right branch positive and decreasing. Following points at known distances from the pole separates a large function value from a genuinely undefined input. Let f(x)=3.5/(x−(1)), with x≠1. Evaluate at x=3.5, exactly 2.5 units to the right of the pole. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f(x)=a/(x−b); f′(x)=−a/(x−b)²
03 · Reflect and transfer
Explain what changes and why.
Halve the distance to the pole. By what factors do the value and slope magnitude change?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.