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Gradient of the tangent

Calculus PowerPoint on the gradient of the tangent, moving from the slope of a secant line to the limit that gives the slope of a curve at a point, with four worked examples.
Gradient of the tangent preview: 14-slide editable PowerPoint presentation
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Look Inside · 10 pages

Lesson cover titled 'Slope of the tangent line' states the objective of understanding slope at a point.
Opening slide of the calculus lesson
Lesson slide introduces tangent and secant lines and gives the secant slope formula using A and B.
Introduction to tangent and secant lines
Lesson slide derives the secant slope formula [f(a+h) − f(a)]/h using points at x = a and x = a+h.
Secant slope expressed using an increment
Lesson slide begins finding tangent slope from the known secant slope between A(a, f(a)) and B(x, f(x)).
Transition from secant slope to tangent slope
Lesson slide shows B approaching A along a curve as x approaches a, with secant slopes nearing tangent slope.
Secant slopes as point B approaches point A
Lesson slide defines tangent slope as a limit as x approaches a and introduces secant slope using h.
Tangent limit and the increment h
Lesson slide shows h shrinking toward zero as B approaches A, making secant slope approximate tangent slope.
Shrinking h to approach the tangent slope
Lesson slide compares tangent slope limits using x approaching a and h approaching zero beside curve diagrams.
Two equivalent limit forms for tangent slope
Lesson slide formally defines the tangent at (a, f(a)) using two slope limits, provided the limits exist.
Formal definition of the tangent line
Worked lesson example finds the tangent to f(x) = x² + 1 at x = 3: slope 6, point (3, 10), and y = 6x − 8.
Example 1: finding a tangent-line equation
Grades
Grade 12, High School
File type
Microsoft PowerPoint
Use
Digital activity

Updated Oct 26, 2026

Subject
Math, Calculus
Topic
calculus, gradient of tangent
Resource types
Teacher Tools, Lesson Plans
Use
Digital activity

About this resource

What's inside this resource

PowerPoint presentation, 14 slides explaining to understand the meaning of the slope of a curve at a point.

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In the classroom

Teaching tips & learning objectives

Learning objectives

  • Find the slope of a tangent line as the limit of secant slopes
  • Write the equation of a tangent line to a curve at a given point

Skills covered

  • Limit calculation — evaluating the limit of secant slopes in the worked examples

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Resource details

Concepts & topics

Core concepts

  • Secant line
  • Tangent line
  • Slope of a curve
  • Limit definition of the derivative

Topics

calculus gradient of tangent slope of a curve

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Categories this resource belongs to

Math Lesson Plans Addition and Subtraction Lesson Plans Algebra Lesson Plans Decimals Lesson Plans Early Math Lesson Plans

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