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# Exam 2 Study Guide APPM 1350

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## Popular in Calculus 1 for Engineers

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This 14 page Study Guide was uploaded by Michael Montella on Tuesday October 20, 2015. The Study Guide belongs to APPM 1350 at University of Colorado taught by Murray Cox in Fall 2015. Since its upload, it has received 49 views. For similar materials see Calculus 1 for Engineers in Applied Mathematics at University of Colorado.

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Date Created: 10/20/15

APPM 1350 Exam 2 FALL SEMESTER 2015 INSTRUCTOR DR Murray ox Chain Rule zd x jimz ma iii in I I f nimsrx m 1 mm rm Example 2 c s m m fr 39 EGEIEEEEEEGHEEDZIH E ij lim I 3 i caimn 2r Si fmn j SEGEIE f 33 CHSIE c s nim j 32 Sim mnix ref SEEE E EE Chain Rule Practice Problem fi ixt3m324 Find fix Solution fig 32122 2313 361 Implicit Differentiation Example Problems m3yEE 2 21 y r11 1 yd 1E1 Eyi y 392I w 131 EU Si li fI y yg fi 653 y EEK Si i f l g yyrl newgamma 3mm yEamnzmi2ywm m yi SLEEI y n yyic 3m yimnimi c s m i y fm r f yi yma m J y i iwimi msim y liif y E y i 39 yESinimja WSW Implicit Differentiation Practice Problem Derive 5E3 33 my Solution SEE r y yiz fmyH y y nyE m Ey Em e 333y 3yi my y y 7342251339 i J 2 2y 12 Ey y rst y 2 Evy 33 3 J y Related Rates 0 Rate of change of one quantity in terms of the rate of change of another Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm3s How fast is the radius ofthe balloon increasing when the diameter is 50cm Kn own U n kn own it w orl iii sir li r all i 3 It 3 rill iiiradii I It sir i1 JET air 1 i 2 2 E It viar2 it It viar2 i 1 a I 395 25w 3 Related Rates practice problem A man walks along a straight path at a speed of 4fts A searchlight is located on the ground 20ft from the path and is kept focused on the man At what rate is the search light rotating when the man is 15ft from the point on the path closest to the search light Hint Draw a picture ofthe situation Linearization 0 By taking linear approximations we can calculate values without a calculator Example To do this without a calculator we must first define a function to use Elfiwll We then define a function to be the linear approximation at x 4 Lit fi4if E41iI 4l Using equation 1 we get dj Using equation 2 we get lial When approximating at x 436 we use Lfmjthe function Ma a fiailj it ltdjrata at l11 Lil5136 2 1 313 H1 4136 2 3 Lima 2m If you would like a better explanation of this particular problem visit Khan Academy That s where I got the problem Tawna uvwhm H pk Minimum and Maximum fc is the absolute maximum of fx on D if Forallxin D fc is the absolute minimum of fx on D if sfwivme fc is the local maximum of fx on D if Absolute 5quot in Maximum Boundary Point Boundary Point ii E forX near C fc is the local minimum of fx on D if E forX near C Local Minimum 33 L lzrelui Minimum 0 There can be multiple global minimums because they can all equal each other 0 Maximum and Minimum will most likely be where the derivative is 0 0 Critical points are where f x 0 or DNE Example IfE3 3E1 Egigi figs 2 41 fiaiiziiarim E 322 II 3 3 i 1 IIIEi 3 Absolute Minimum 2 3 It4i Absolute Maximum 4 17 f x0whenxisOor2 Mean Value Theorem Roll s theorem f fX is continuous on ab fX is differentiable on a b and fa fb Then Elm391abjfej i There exists a value c in a b such that f c 0 a c 1 Mean Value Theorem Lagrange if fX is continuous on a b and fX is differentiable on a b fiiiimfiili Instantaneous Average ill til Then 33 liniaibifji SEE nt Tangent t 10 Example f m m DE flsz 3I2 1 W EE E J m39m A W ll Increasing and Decreasing f x gt O on a b gt fx is increasing on a b f x lt O on a b gt fx is decreasing or a b Concavity and lnflection Points fquotc gt 0 for graphs concave UP f39C 0 fquotc gt 0 gt Local Minimum fquotc lt O for graphs concave DOWN f39C 0 fquotc lt 0 gt Local MaXimum y34 43135 isquot E m 1 m mgr 1 critical points is l i mE EE E Wm Concave Down Ei 2l Concave Up lZ ENE 1 l Concave Up Sketching Curves Optimization Understand the problem Draw a diagram Add meaningful notation Build a function with multiple values QX y Turn it into a one variable function QX 991WIV Optimize You have a 2400 foot long rope What is the largest area you can create next to a wall I Ali ail 1 THE1 Domainhl W W Eu l 2 21130 IE Mlil 2m Afw 1 mf llillll Ew LU m2 A rw 1 24m lw 2 Am ll 23 film Af T E g j w a El 1200 2 El Max Area 720000 sq ft w 600ft l 1200ft l4

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