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Calculus

Find the equation of a tangent line

Use the derivative to get the slope at a point, then build the line y = m(x − a) + f(a).

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A tangent line touches the curve at one point and has the same slope as the curve there. The slope is the derivative.

Take f(x) = x² at x = 1. The point is (1, 1) and the slope is 2, so the line is y = 2(x − 1) + 1 = 2x − 1.

Classic CAS

Start on the main screen: Press home to open the Home menu (Calculator, Graphs, Settings, and more). Choose Calculator, or press esc to go back to the calculation page.

  1. Type expand( then the point-slope form: the derivative at x = 1 times (x − 1), plus the y value 1. Press enter.

    expand(d(x^2▶,x=1)×(x−1)+1)enter

You should see2·x-1

Classic CAS key reference · Classic CAS manual

Classic Graphing 84

Start on the main screen: Press 2nd then mode (QUIT) to get back to the home screen. Pressing clear on a menu also backs out of it.

  1. First get the slope. MATH 8 is nDeriv: type X², X, 1 and close the bracket.

    math8X,T,θ,nx²,X,T,θ,n,1)enter

  2. Now find the intercept b = y − m·x. The slope is stored in Ans (2nd, then the (−) key). Type 1 − Ans × 1.

    1−2ndANS(−)×1enter

You should seeThe slope is 2 and the intercept is -1, so the tangent line is y = 2x − 1.

Classic Graphing 84 key reference · Classic Graphing 84 manual

Natural Color Graphing

Start on the main screen: Press MENU then 1 to open Run-Matrix, the main calculation screen. Press EXIT to back out of a bar or screen, and SHIFT then EXIT (QUIT) to return to the app start.

  1. Open the graph screen, enter X² and press EXE, then draw it with F6.

    MENU3X,θ,Tx²EXEF6

  2. Press F4 (SKETCH), then F2 (tangent). Type 1 for the point and press EXE twice. The line is drawn and its slope is displayed.

    F4F21EXEEXE

You should seedy/dx=2

Natural Color Graphing key reference · Natural Color Graphing manual

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