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{Chapter 4 Practice
AP Calculus
Differentiate:𝑑𝑑𝑥sec𝑥=¿
𝑑𝑑𝑥csc𝑥=¿
𝑑𝑑𝑥tan 𝑥=¿
𝑑𝑑𝑥cot𝑥=¿
𝑑𝑑𝑥sin−1𝑥=¿
𝑑𝑑𝑥sec−1 𝑥=¿
𝑑𝑑𝑥csc−1𝑥=¿
𝑑𝑑𝑥tan− 1𝑥=¿
Differentiate: sec x tan x
-csc x cot x
𝑑𝑑𝑥tan 𝑥=𝑠𝑒𝑐2𝑥
𝑑𝑑𝑥cot𝑥=−𝑐𝑠𝑐2𝑥
𝑑𝑑𝑥sin−1𝑥=
1
√1−𝑥2
𝑑𝑑𝑥csc−1𝑥=−
1
¿𝑥∨√𝑥2−1
𝑑𝑑𝑥tan− 1𝑥=
1
𝑥2+1
To the nearest thousandth, calculate the slope of the tangent where x = 4:
𝑥2−4 𝑦2=4
To the nearest thousandth, calculate the slope of the tangent where x = 4:
𝑥2−4 𝑦2=4Differentiate implicitly:
2 𝑥 ∙𝑑𝑥𝑑𝑥−8 y ∙
𝑑𝑦𝑑𝑥
=0
¿−8 y ∙𝑑𝑦𝑑𝑥
=−2 𝑥
¿−8 y ∙𝑑𝑦𝑑𝑥
=−2𝑥−8 𝑦
→𝑑𝑦𝑑𝑥
=𝑥4 𝑦
Find coordinates of y when x = 4and substitute into dy/dx equation:
h𝑊 𝑒𝑛𝑥=4 , 𝑦=±√3𝑑𝑦𝑑𝑥
=44¿¿
𝑑𝑦𝑑𝑥
=44¿¿
• Be prepared for NO CALCULATOR section!• Basic chain rule, product rule, quotient rule• Basic trig derivatives• Inverse trig derivatives• Implicit differentiation • Use limits to find values to make a
piecewise function differentiable (and continuous).
• Related Rates
Ch. 4 Test Review Topics
Derivatives Practice
𝑑𝑑𝑥
𝑐𝑜𝑠4(5 𝑥−4 )
𝑑𝑑𝑥
7𝑐𝑜𝑠5𝑥
𝑑𝑑𝑥x ∙ csc−15𝑥
𝑑𝑑𝑥
𝑒3𝑥
7 𝑥−4
Derivatives Practice
𝑑𝑑𝑥
𝑐𝑜𝑠4 (5𝑥−4 )=−20 (cos (5 𝑥−4 ) )3 ∙ sin (5 𝑥−4 )
= 35 sec 5x tan 5x
=
=
Differentiate implicitly:
4 𝑥2+ tan 𝑥𝑦=𝑦3
Differentiate implicitly:
Derivative:
Useful Related Rates Formulas
𝐶𝑜𝑛𝑒𝑉𝑜𝑙𝑢𝑚𝑒 :𝑉=13𝜋 𝑟2h
𝐶𝑖𝑟𝑐𝑙𝑒 𝐴𝑟𝑒𝑎 :𝜋𝑟2
Cylinder Volume: V =
h𝑃𝑦𝑡 𝑎𝑔𝑜𝑟𝑒𝑎𝑛 h𝑇 𝑒𝑜𝑟𝑒𝑚 :𝑎2+𝑏2=𝑐2
𝑆𝑖𝑚𝑖𝑙𝑎𝑟 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 (𝑆𝑒𝑡 𝑢𝑝𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛)
Ch. 4 R Problems, pg. 180: R4 ad, R5a, R6, R8b, R9 (pretty hard)
Suggested Review
Additional ReviewOnline videos, PPTSExamples from notes4.2 #1-15 odd, 4.3 1-19 odd4.4 1-25 odd, 4.5 13-23 odd