Slope of a Curve: Derivatives
By the end of this course, you will be able to compute the slope of any polynomial curve at a given point using the limit definition of the derivative.
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Start when you are ready to enter this Stage's 11 scenes and explore, respond, and learn as you go.
How do you compute the slope of a curved line at a single point when the standard rise/run formula only works for straight lines?
- secant-slope
- Slope of a line through two points on a curve approximates the instantaneous slope.
- difference-quotient
- The expression (f(x+h)-f(x))/h gives the slope of the secant line.
- limit-to-zero
- As h approaches 0, the secant slope approaches the tangent slope, if the limit exists.
- derivative-definition
- The derivative f'(x) = lim_{h→0} (f(x+h)-f(x))/h is the instantaneous slope function.
- tangent-line
- The tangent line at a point has slope equal to the derivative at that point.
The slope at a point is just the slope of the curve itself, like for a line.
Clarify that slope is defined only for lines; for curves we use the tangent line slope.
The derivative gives the slope of the entire curve, not just at one point.
Emphasize that derivative is a function that gives the slope at each x, but at a single x it's one number.
Taking h=0 directly in the difference quotient gives the exact slope.
Show that h=0 leads to division by zero; avoid that by taking the limit.
- slope formula (y2-y1)/(x2-x1)
- function notation
- basic idea of a limit
- derivative rules (power rule, etc.)
- chain rule
- implicit differentiation
- applications to physics
- Compute the derivative of a quadratic function at a given point using the limit definition.
- Explain why the limit of the secant slope equals the tangent slope.
- Find the equation of the tangent line to a curve at a point.
- Use the limit concept to understand instantaneous rate of change in other contexts like velocity.
A student comfortable with algebra and the concept of slope of a straight line.
- 01The Problem: Slope of a Curved LineslideOrientationObserve
Introduce the challenge of finding the slope at a single point on a curve, motivating the need for a new approach.
- Straight line slope = rise/run is easy
- Curves bend – slope changes at every point
- We need a method to find slope at one exact point
- 02Guess the SlopequizPredictionPredict
Ask the student to predict the slope of y = x² at x = 1, surfacing the common misconception.
- Make a prediction before learning the method
- Options: 0, 1, 2, don't know
- 03Approximation: The Secant LineslideModel buildingObserve
Show how drawing a line through two nearby points (secant) gives an approximate slope.
- Secant line connects two points on the curve
- Slope = (f(x+h)-f(x))/h
- As points get closer, approximation improves
- 04Watch the Secant Become TangentinteractiveModel buildingConstruct
Adjust a slider to make h smaller and see the secant line approach the tangent line.
- Move the h slider to see secant lines for different distances
- Observe how the slope stabilises as h→0
- The limiting line is the tangent
- 05The Limit Definition of DerivativeslideModel buildingObserve
Present the formal definition f'(x) = lim_{h→0} (f(x+h)-f(x))/h as the instantaneous slope.
- Derivative f'(x) = limit of secant slopes
- Notation: f'(x), dy/dx
- The limit as h→0 avoids division by zero
- 06Worked Example: f(x) = x² at x = 1slideModel buildingObserve
Step-by-step calculation of derivative using limit definition, showing the derivative function and then evaluating at x=1.
- Write difference quotient: ((1+h)² - 1²)/h
- Simplify to 2 + h
- Take limit h→0 → slope = 2
- 07Practice the LimitquizPracticeApply
Compute the derivative of f(x) = x²+1 at x = 2 using the same method, choosing the correct numeric slope.
- Apply the limit definition step by step
- Remember that h→0, not h=0
- 08From Slope to Tangent LineslideApplicationObserve
Once we have the derivative slope at a point, we can write the equation of the tangent line using point-slope form.
- Tangent line: y - f(a) = f'(a)(x - a)
- Use the derivative value as slope m
- Example: for f(x)=x² at x=1, tangent: y-1 = 2(x-1)
- 09Write the Tangent LinequizApplicationApply
Given the derivative and a point, find the equation of the tangent line (short answer).
- First confirm derivative value
- Then apply point-slope formula
- 10Putting It All TogetherslideSynthesisExplain
Summarise the journey from secant to derivative to tangent, reinforcing the key steps.
- Secant slope → difference quotient → limit → derivative
- Derivative gives slope at one point
- Tangent line uses that slope
- 11Final CheckquizAssessmentApply
Assessment: compute derivative and tangent line for a new curve to confirm mastery.
- Compute derivative using limit definition
- Find equation of tangent line at given point
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