L17 the differentials (applications)
- 2. EXAMPLE 1: Use differentials to approximate the change in the area of a square if the length of its side increases from 6 cm to 6.23 cm. Let x = length of the side of the square. The area may be expressed as a function of x, where A= x2 . The differential dA is ( ) dxx2dAdxx'fdA ⋅=⇒⋅= Because x is increasing from 6 to 6.23, you find that Δ x = dx = .23 cm; hence, ( )( ) 2 cm76.2dA cm23.0cm62dA = = ( ) .cm2.8129isyareainincrease exactthethatNote6.23.to6fromincreaseslengthsideitsas cm2.76elyapproximatbyincreasewillsquaretheofareaThe 2 2 ∆
- 3. EXAMPLE 2: Use the local linear approximation to estimate the value of to the nearest thousandth. 3 55.26 ( ) ( ) ( ) ( ) ( ) 2.9830.0167-3 55.26 60 1 355.26 therefore ;327 that less 60 1 tely approxima be will 55.26 that implies which 0167.0 60 1 100 45 27 1 45.0 273 1 dy , Hence 0.45dxx then 26.55, to 27 from decreasing is x Because dx x3 1 dx x 3 1 dy xxf dx xf'dy is dy aldifferenti The 27.x namely 26.55, to close relatively is and cube perfect a is that xof value convenient a choose ,xxf is applying are you function the Because 3 33 3 3 2 3 2 3 2 3 1 3 ≈≈ −≈= −=−=−⋅=−⋅= −==∆ == =→= = = −
- 4. EXAMPLE 3: If y = x3 + 2x2 – 3, find the approximate value of y when x = 2.01. 2.xof value original an to 0.01dxxof increment an applyingof result the as 2.01 gconsiderin are we then 0.01,22.01 write weif that Note dy. y for solve shall we then value, eapproximat the find to asked simply are we since but y y is value exact The ===∆ +=+ ∆+ ( ) ( )( ) 20.1320.013dyy is ionapproximat required the,therefore 20.001.0812dy then ,01.0dx and 2x when and 13388y then ,2x when dxx4x3dy then 3x2xy Since 2 23 =+=+ =+= == =−+== += −+=
- 5. EXAMPLE 4: Use an appropriate local linear approximation to estimate the value of cos 310 . ( ) ( ) ( ) ( ) ( )( ) ( ) 8573.0008725.0866.0dyy is ionapproximat required the,therefore 008725.001745.05.0dy 180 130 sindy then ,01745.0 180 1dx and 30x when and 866.030cosy then ,30x when dx x sindy then x cosy Let 0 00 0 00 00 =−+=+ −=−= π •−= = π == === −= =
- 6. EXAMPLE 4: Use an appropriate local linear approximation to estimate the value of cos 310 . ( ) ( ) ( ) ( ) ( )( ) ( ) 8573.0008725.0866.0dyy is ionapproximat required the,therefore 008725.001745.05.0dy 180 130 sindy then ,01745.0 180 1dx and 30x when and 866.030cosy then ,30x when dx x sindy then x cosy Let 0 00 0 00 00 =−+=+ −=−= π •−= = π == === −= =