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    • ▾Calculus 1
      • ▸Limits and continuity
        • •Intro to calc 1
        • •Estimating limits from graphs
        • •Estimating limits from tables
        • •Intro to continuity
        • •Finding and classifying discontinuities
        • •Finding the value that makes the function continuous
      • ▾Derivatives: definition and basic derivative rules
        • •Proving differentiable implies continuous
        • •Derivative of a function from first principles
        • •Finding the tangent or normal line through the given point
        • •Constant rule
        • •Constant multiple rule for derivatives
        • •Sum, difference, and power rules for derivatives
        • •Product rule
        • •Proving the product rule
        • •Quotient rule
        • •Derivatives of trig functions
        • •Proving properties of even and odd functions (calculus 1)
        • •Leibniz's derivative notation
        • •Logarithmic functions
      • ▸Derivatives: composite, implicit, and inverse functions
        • •Derivative at a point from first principles
        • •Derivatives of exponential functions
        • •Derivatives of logarithmic functions
        • ▸Derivatives of hyperbolic and inverse hyperbolic functions
          • •Using the derivatives of the hyperbolic functions
          • •Proving the derivatives of the hyperbolic functions
          • •Proving the derivatives of the inverse hyperbolic functions
        • •Derivatives of inverse trig functions
        • •Derivatives of transcendental functions
        • •Derivatives of trigonometric functions under transformations
        • •Derivative using u-substitution
        • •Determining derivatives from graphs
        • •Higher order derivatives
        • •Chain rule
        • •Derivatives of inverse functions
        • •Choosing the derivative rules to use
      • ▸Applying derivatives to analyze functions
        • •Applications of trigonometric derivatives
        • •Extreme value theorem
        • •First derivative test
        • •Mean value theorem
        • ▸Monotonic intervals and functions
          • •Intro
          • •Proofs
        • •Related rates
        • •Second derivative test
        • •Concavity
        • •Factoring a cubic polynomial with a double root
        • •Graphing using derivatives
        • ▸Special points
          • ▸Finding critical values from graphs
            • •Lesson
            • •Practice
          • •Finding critical values using derivatives
          • ▸Inflection points
            • •Graphically finding points of inflection
            • •Using the second derivative to find points of inflection
          • •Classifying stationary points
          • •Sketching polynomials using stationary points
        • •Bisection method
        • •Newton's method
        • •False position method
        • •L'Hopital's rule
      • ▸Integrals
        • •Basic integration problems
        • ▸Integration by u-substitution
          • ▸Integration by u-substitution (less difficult)
            • •Lesson
            • •Practice
          • •Integration by u-substitution (more difficult)
          • •Integrating exponential functions by u-substitution
        • •Visually determining antiderivative
      • ▸Applications of integrals
        • •Proving the formula for the area of a circle
        • •Area between two curves
        • ▸Solids of revolution
          • •Disc and washer methods (circular cross sections)
          • •Volumes of solids with known cross sections
          • •Shell method
          • •Volume of a sphere (calculus \(2\))
          • •Volume of a cone (calculus 2)
     › Calculus 1 › Derivatives: definition and basic derivative rules

    Finding the tangent or normal line through the given point

    Students will learn how to find the tangent and normal line, given an equation and a point.

    Equations of Tangents & Normals by Eddie Woo

    Conclude by giving your students these challenges:

    • There and Back by NRICH
    • 2000 AMC 8, Problem 22
    • Areas of Nested Squares

    Solve \(x^2 + 19x - x! = 0.\)

    Here's one solution. And here's another:

    $$\begin{align} & x^2 + 19x - x! = 0 \\ & x^2 + 19x = x! \\ & x + 19 = (x - 1)! \\ & (x - 1) + 20 = (x - 1)! \\ & 1 + 20/(x - 1) = (x - 2)! \end{align}$$

    The right-hand side is an integer, so the left-hand side must also be an integer. Thus, \(x - 1\) must divide \(20.\) So \(x - 1\) must be \(1,\,2,\,4,\,5,\,10,\) or \(20.\) Thus, \(x\) must be \(2,\,3,\,5,\,6,\,11,\) Testing each value, we find \(x = 5\) is the only value that works.

    Find the radius of the circle:

    Prereqs include knowing a rectangle has four right angles, knowing the intersecting chords theorem, and knowing the Pythagorean theorem. Here's the solution. Before watching this video, you should be aware it contains one mistake. He uses Thales's theorem to conclude AE is a diameter, but Thales's theorem can only be used to prove an angle is \(90^\circ,\) subject to certain conditions. In this case, we know AE is a diameter because ABE is a right angle. If it's not obvious, you could imagine copying the path ABE, rotating it 180 deg around the center of the circle. What you get is a rectangle circumscribed by the circle. Thus AE is the diagonal of the rectangle, and O must be its midpoint, as any rectangle and its circumcircle have the same center, by symmetry.

    Practice problems and additional lessons