Review Sheet for MATH 124 at UA
Review Sheet for MATH 124 at UA
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Date Created: 02/06/15
A Chapter 3 Shortcuts to Differentiation Section 37 Implicit Functions Section 38 Hyperbolic Functions Section 39 Linear Approximation Math 124 Section 023 Fall 2007 Instructor Paul Dostert p1 A 3 7 Implicit Functions A function y f2 is called explicit since it is written as y in terms of 23 An equation such as f1 y c where c is a constant is called an implicit function An example of an implicit function is the circle 22 y2 1 How do we obtain an 6amp7 expression for do We simply take the derivative of the whole expression with respect to 23 i 2 2 i dxlt y da1 The right side goes to zero and we use the chain rule on the left d 2332y zO Solving for dydaj we get I daj y p2 A 3 7 Implicit Functions Ex Differentiate each of the following a Ijlny 1 b 32 gay 2 2 c oos2 siny 0 d Ijm y1 2I 2 e a siny oos2y oosy f ij ootay Ex Find the equation of the tangent line to the curve with equation 25 myZ y3 4 at the point 1 2 Ex Prove the curves 22 y2 5 and 4232 9y2 2 72 are orthogonal Show the tangent lines are perpendicular at every point of intersection p3 A 38 Hyperbolic Functions We define cosha and sinha by 69 6 9 em 6 9 cosha T smha tanh a is defined as expected sinh a 69 6 9 tanha cosha 69 6 9 Many properties of trig functions hold also for hyperbolic functions Calculate each of the following a cosh O b sinh O c cosh cl sinh e cosh 132 sinh 132 23 23 p4 A 38 Derivatives of Hyperbolic Functio Find each of the following d E sinha d cosha 2 d9 d tanha 2 Ex Simplify sinhlna Ex Find each of the following derivatives a y cosht sinh2t b f6 lnoosh2 9 sinhsinha2 6 9 v p5 A 39 Linear Approximation Look at any differentiable function on a small interval What does it look like For example what does Sina look like for 1 lt a lt 1 Suppose we want to appoximate a differentiable function f2 at a point a a by a linear function How can we do this What equation do we get The Tangent Line Approximation Suppose f is differentiable at d Then for values of a near a the tangent line approximation to f 23 is given by f W faf39a a This expression fa f a2 a is called the local linearization of f near a a The error of the linear approximation is fl a a a2 p6 A 39 Linear Approximation An Aside What if a linear approximation isn t good enough Can I use a quadratic f W fa f a a f a 602 What if a quadratic approximation isn t good enough Taylor Series Approximation Suppose f is a function with infinitely many derivatives For any point a a we have me Z f av ar Notice this is and not w The Taylor series is one extremely important example of how derivatives can be used in practice p7 A 39 Linear Approximation Ex Find the linear approximation to fv 1 1 Qatle a Ex What is the local linearization of 6quot near a 1 Ex The acceleration due to gravity 9 is given by GM 9 T2 where M is the mass of the earth r is the distance from the center of the earth and G is the universal gravitational constant Show that when r changes by Ar the change in the acceleration due to gravity Ag is given by Ag Zg 7 p8 A Chapter 3 Review Questions Ex Are the following statements true or false Explain a The 100th derivative Of sinh 2 is cosh 23 b If f23 is defined for all 23 then f 23 is defined for all 23 c If f23 is increasing then f 23 is increasing d fg 23 is never equal to f 23g 23 e The only functions whose fourth derivatives are equal to cost are of the form cost C where C is any constant f If f and g are concave up for all 23 then f3 g23 is concave up for all 23 g If f and g are concave up for all 23 then f2g2 is concave up for all 23 h If f and g are concave up for all 23 then f3 g23 cannot be concave up for all 23 i If f and g are concave up for all 23 then fg2 is concave up for all 23 p9
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