inversesanddifferentiability

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New York University *

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MISC

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Mathematics

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Nov 24, 2024

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pdf

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21

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September 17, 2021 Inverses and differentiability BC Calc 2021 4.3 I know the relationship between inverse functions and their derivatives and can derive all 6 inverse trig functions’ derivatives . 4.4 I understand the meaning of differentiability and can explain its relationship to continuity
September 17, 2021 !"# % &’()*+ ,-".& , /.)0&(") ,)1 (&+ ()234+35 1. Inverse functions "undo" each other 2. They are reflections over the line y=x 3. Their domains (x) and ranges (y) are reversed 4. To find an inverse, switch x and y and solve for y. 5. All functions have inverses, but only one-to-one functions have inverse functions.
September 17, 2021 Let's use this graph to remind ourselves about the relationship between inverse functions
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September 17, 2021 Let's generalize! Now write one statement relating the two derivatives:
September 17, 2021 f(x) f -1 (x) (a,b) (b,a) (f -1 )'(b) = 1/f '(a)
September 17, 2021 Formally: "the output of f inverse"
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September 17, 2021 Try this one again (most people missed it)
September 17, 2021 Can we use this formula to derive the derivative of inverse tangent? d/dx tan -1 (x) =
September 17, 2021 Derivatives of Inverse Trigonometric Functions
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September 17, 2021 Why abs? Here is a graph of secant, with its restricted domain in green. Note that sec -1 (x) is always increasing
September 17, 2021
September 17, 2021 !’3 67(& 8(*) 94"-:3; !’3 -,+3 "/ , <= /&> &,:: 37(& +(*) (+ ?= /33& ,-"23 &’3 14(234@+ 3A3 :323:> B’3) 0,4+ ,43 /,4 ,C,AD &’3 +(*) (+ ’,41 &" 43,1 -30,.+3 "/ &’3 1(+&,)03> B’3) &’3A ,43 234A 0:"+3D &’3 +(*) (+ ’,41 &" 43,1 -30,.+3 14(234+ ’,23 &" :""E .# ,& , +&33# ,)*:3> !’3 +(*) (+ 3,+(3+& &" 43,1 C’3) &’3 1(+&,)03 7 (+ +.0’ &’,& &’3 ,)*:3 ࠵? ,& &’3 14(234@+ 3A3 (+ ,+ :,4*3 ,+ #"++(-:3> Write 2 trig equations. Use algebra and substitution to isolate ࠵? as a function of x only (no a).
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September 17, 2021 20 ft 30 ft x a Goal: maximize ࠵? Non-wimpy strategy: Find tan(a)=30/x tan(a+ Ɵ )=50/x
September 17, 2021 If car is 38.729 feet away, the angle will be maximized. Theta is = ____
September 17, 2021 #26. The Exit sign problem. The maximum is where x ≈ 38.72983349
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September 17, 2021 F(//343)&(,-(:(&A G / @H0I 37(+&+ (/G ,)1 /H7I (+ 0")&().".+ ,& 0> ' '
September 17, 2021 The other big connection from the Desmos lesson was .... 1) If a function is differentiable (has a derivative) it must be continuous 2) If a function is continuous, it might not be differentiable. "smooth! Therefore no jumps or holes" "continuous, but might have a cusp"
September 17, 2021 outputs = at 3? Slopes equal @ x = 3? Differentitable at x = 3?
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September 17, 2021 So: y=csc -1 (x) cscy=x
September 17, 2021 y = sinx vs y = sin -1 x angle angle trig ratio trig ratio J)234+3 !4(* K.)0&(")+ L,)*3 (+ 43+&4(0&31 &" ;,E3 +() MN 7 , /.)0&(") newsflash: "the reciprocal of sine" inverse sine "un-sine"
September 17, 2021 1. y = sinx, y = sin -1 x restricted domain: output: 2. y = cosx y = cos -1 x 3. y = tanx, y = tan -1 , J)234+3 !4(* K.)0&(")+ restricted domain: output: restricted domain output:
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