Review 3 log diff 12.3 3.4 4 PDF

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1) Find a) b) c) d) e) f) g) h)

dy dx 4 2 y= ( 2 x−sin ( 3 x ) ) x y= ( 3 x +2)√ √ 2 x +1 y =tan(2 x )+ln 2 2 ( x +1 ) x y= ( 2 x )tan ( ) x 2 (¿csc x)+ ln(sec x ) y=log 2 ¿ 2 ( 7 x +2 ) y= Use logaritmic differentiation ( 2 x3 +1 ) √ 5 x−1 2 2 y cos x + x cos y=π 3 cos x −1 =2+3 x 1+cos y

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2) Find the higher order derivative. a) f ' ' ( x ) if f (x)=ln ( sin 3 x ) b) f 34 (x )if f ( x )=cos ( 9−3 x ) +2 x 33 3) The profit P made by a cinema from selling x bags of popcorn can be modeled by C=0.6 x+ 7500, 0 ≤ x ≤ 50000 1 ( 65000 x − x 2) , 0 ≤ x ≤50000 R= 20000 a) Write the profit function for this situation. b) Determine the intervals on which the profit function is increasing and decreasing. c) Find the marginal profit function P' ( x ) . d) Compute P' (20000) and interpret your result. 4) The demand and cost functions for a product can be modeled by p=211−0.002 x C=30 x+1 500 000 and where x is the number of units produced. a) What is the marginal revenue when 80 000 units are produced? Interpret your result. b) What is the marginal profit when 80 000 units are produced? Interpret your result.

x=

12 p p−5

where p is the price per unit when x 5) The demand function for a product is units are demanded. a) Find the elasticity of demand function E(p). b) If the price of the product at $25 decreases by 4%, what is the approximate percentage change in demand?

c) Will the total revenue increase, decrease, or remain constant at p = $25? 6) The demand function for a product is given by unit when x units are demanded. a) Find the elasticity of demand function E(p). b) What price would maximize revenue?

x=

√

p p−1

where p is the price per

7) Find the critical numbers and the open intervals on which the function is increasing or decreasing. a) f ( x )=3 x 3 + 12 x 2 + 15 x −2 x2 +3 x −8 b) f ( x )= x 2x c) f ( x )= 16− x2 8) Find all relative extrema of the function. a) f ( x )=2 ( x−3 )3 2 x 2 +7 x +8 b) f ( x )= x +2 1 4 5 3 2 c) f ( x )= x − x +2 x +3 3 4 9) Find the absolute extrema of the function −x 3−4 on [1,4 ] a) f ( x )= 2 x 5 4 20 3 b) f ( x )= x − x +6 on[− 1,3 ] 3 2 10) Find the inflection points and the open intervals on which the function is concave up or concave down. a) b)

3

f ( x )= ( x−5 ) 2 −x f ( x )= 2 x −4

11) Use the second derivative test to find the relative (local) extrema of 1 4 5 3 2 35 a) f ( x )= x − x +2 x + 3 3 4 2 b) f ( x )= ( x 2−1 )...