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;SPMlt;P;SPMgt; Chapter 3: Worksheet 3 Jack K. Cohen Colorado School of Mines

;SPMlt;P;SPMgt; ;SPMlt;BR;SPMgt; ;SPMlt;BR;SPMgt; ;SPMlt;BR;SPMgt; Techniques of Differentiation: Product and Quotient Rules

;SPMlt;P;SPMgt; Suggested Problems Section 3.2: See below.

;SPMlt;P;SPMgt; ;SPMlt;BR;SPMgt; ;SPMlt;BR;SPMgt; ;SPMlt;BR;SPMgt;

;SPMlt;P;SPMgt; For the next few days, your job is get some mechanics well in hand. It is not practical to use a computer for every routine calculation! However, in the Colorado School of Mines calculus environment, you can make your learning more efficient by getting instant feedback on the accuracy of hand calculation of derivatives.

;SPMlt;P;SPMgt; ;SPMlt;PRE;SPMgt; (* Take a simple derivative with Mma *) D[x^3 + 3x, x] 2 3 + 3 x

(* Put in problem 3.2.25, but change variable to ;SPMquot;t;SPMquot; *) g[t_] := (1/t - 2/t^2) / (2/t^3 - 3/t^4) D[g[t], t] (* Ask for derivative *) 4 -2 12 6 -2 1 -- - t (-- - --) (-- + -) 3 5 4 2 t t t t t -------- - ------------------ -3 2 -3 2 2 -- + -- (-- + --) 4 3 4 3 t t t t

(* What a mess! Fortunately Mma has a Simplify function. The symbol 1 Simplify[2 2 t (12 - 13 t + 4 t ) -------------------- 2 (-3 + 2 t)

(* If we anticipate the need for Simplify, we can do it all in one step. Reminder: g[] has been defined as a true function, so we can change the variable to x if we like *) Simplify[ D[g[x], x] ] 2 x (12 - 13 x + 4 x ) -------------------- 2 (-3 + 2 x)

(* Mma also understands the prime notation for derivative. *) Simplify[ g'[t] ] 2 t (12 - 13 t + 4 t ) -------------------- 2 (-3 + 2 t) ;SPMlt;/PRE;SPMgt;

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;SPMlt;OL;SPMgt; ;SPMlt;LI;SPMgt;Do the even numbered problems (2-50) in section 3.2. Be sure to vary your notation between ;SPMlt;!-- MATH #math3##tex2html_wrap_inline110#,#tex2html_wrap_inline111#, f'(x) --;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 2.10ex; vertical-align: 176.33ex; ;SPMquot; SRC=;SPMquot;img1.png;SPMquot; ALT=;SPMquot;#math4##tex2html_wrap_inline113#;SPMquot;;SPMgt;,;SPMlt;IMG STYLE=;SPMquot;height: 3.08ex; vertical-align: 175.42ex; ;SPMquot; SRC=;SPMquot;img2.png;SPMquot; ALT=;SPMquot;#math5##tex2html_wrap_inline115#;SPMquot;;SPMgt;, ;SPMlt;I;SPMgt;f';SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;) and ;SPMlt;I;SPMgt;Df;SPMlt;/I;SPMgt; (;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;). Make sure that there is no sense of panic when the independent variable is (say) ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; instead of ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;. Check your answers using D[f[x], x], f'[x], Simplify (answers will not be given for this exercise, ask your instructor about anything that stumps you) . Dispensation: if you get 3 in a row correct with no help from anyone or from you can skip to harder looking exercises.

;SPMlt;P;SPMgt; ;SPMlt;/LI;SPMgt; ;SPMlt;LI;SPMgt; According to the inverse square law, the weight of an object near the Earth is given by ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math6#

W = #tex2html_wrap_indisplay117#

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;I;SPMgt;W;SPMlt;/I;SPMgt; = ;SPMlt;IMG STYLE=;SPMquot;height: 3.08ex; vertical-align: 174.63ex; ;SPMquot; SRC=;SPMquot;img3.png;SPMquot; ALT=;SPMquot;#math7##tex2html_wrap_indisplay119#;SPMquot;;SPMgt; ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; where ;SPMlt;I;SPMgt;G;SPMlt;/I;SPMgt; is the universal constant of gravitation, ;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt; is the mass of the Earth, ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; is the mass of the object and ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; is the distance from the Earth's center. For problems concerning objects near the Earth's surface, it is convenient to introduce the surface gravitational constant ;SPMlt;!-- MATH #math8#g = GM/R2 --;SPMgt; ;SPMlt;I;SPMgt;g;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;GM;SPMlt;/I;SPMgt;/;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt; where ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; is the mean radius of the Earth. This lets us write the inverse square law as ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math9#

W = mg#tex2html_wrap_indisplay122##tex2html_wrap_indisplay123##tex2html_wrap_indisplay124#.

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;I;SPMgt;W;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;mg;SPMlt;/I;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 4.97ex; vertical-align: 172.20ex; ;SPMquot; SRC=;SPMquot;img4.png;SPMquot; ALT=;SPMquot;#math10##tex2html_wrap_indisplay126##tex2html_wrap_indisplay127##tex2html_wrap_indisplay128#;SPMquot;;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 5.67ex; vertical-align: 172.04ex; ;SPMquot; SRC=;SPMquot;img5.png;SPMquot; ALT=;SPMquot;#math11##tex2html_wrap_indisplay130#;SPMquot;;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 4.90ex; vertical-align: 172.08ex; ;SPMquot; SRC=;SPMquot;img6.png;SPMquot; ALT=;SPMquot;#math12##tex2html_wrap_indisplay132##tex2html_wrap_indisplay133##tex2html_wrap_indisplay134##tex2html_wrap_indisplay135#;SPMquot;;SPMgt;. ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;

;SPMlt;OL;SPMgt; ;SPMlt;LI;SPMgt;What is the rate of change of ;SPMlt;I;SPMgt;W;SPMlt;/I;SPMgt; with respect to ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt;? ;SPMlt;/LI;SPMgt; ;SPMlt;LI;SPMgt;Evaluate ;SPMlt;I;SPMgt;W;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt;) and ;SPMlt;I;SPMgt;W';SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt;) when ;SPMlt;!-- MATH #math13#r = R = 3960 --;SPMgt; ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; = 3960 miles and ;SPMlt;I;SPMgt;g;SPMlt;/I;SPMgt; = 32 ft/sec;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt; for a mass of 5.5 pounds. ;SPMlt;/LI;SPMgt; ;SPMlt;LI;SPMgt;Near the surface of the Earth, the approximation ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; is commonly used to give ;SPMlt;!-- MATH #math14#W #tex2html_wrap_inline138# mg --;SPMgt; ;SPMlt;I;SPMgt;W;SPMlt;/I;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 6.15ex; vertical-align: 173.15ex; ;SPMquot; SRC=;SPMquot;img7.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline140#;SPMquot;;SPMgt; ;SPMlt;I;SPMgt;mg;SPMlt;/I;SPMgt;. Redo part (b) with this approximation and comment on its validity. ;SPMlt;/LI;SPMgt; ;SPMlt;LI;SPMgt;Repeat the last part, but with ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; replaced by 5500 miles.

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