Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous Problem set0/10 Problem set 20/10 Practice∞

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Inverse Functions and the Reflection Across y = x

Why is the graph of an inverse the original reflected across y = x, and what turns the tangent curve into an arctangent that levels off?

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Your summary note

1. The inverse $g = f^{-1}$ and $g(f(x)) = x$
  Write the definition: a one-to-one $g(f(x)) = x$ makes $g(y) = x$ a function with $g(f(x)) = x$ , and carry $f(x) = \sqrt{x}$ through to $g(y) = y^2$ .
2. The inverse graph as a reflection in the diagonal
  Record that graphing the inverse on the same axes trades each point $(x, y)$ for $(y, x)$ , a reflection in the diagonal, and draw $\sqrt{x}$ against $x^2$ , $x \ge 0$ .
3. The arctangent graph and its asymptotes
  State that $y = \arctan x$ means $\tan y = x$ , reflect the tangent on $(-\pi/2, \pi/2)$ , and record the horizontal asymptotes $y = \pm\pi/2$ with the limits at $\pm\infty$ .
Attempt 1 of 2