Linear Approximation
& Error Estimation
Miscellaneous on-line topics for
Calculus Applied to the Real World
Exercises

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Note: All numerical answers entered must be accurate to $4$ decimal places; otherwise they will be marked wrong.

Find the linear approximation near the indicated value for each of the following functions.

L1. $f(x) = 3x + 5,$ $a = 2$ $L(x) =$    
L2. $f(x) = 3x^2 - 4x + 5,$ $a = - 1$     $L(x) =$    
L3. $f(x) = \frac{x}{2x+1},$ $a = 0$ $L(x) =$    
L4. $f(x) = e^x,$ $a = 0$     $L(x) =$    
L5. $f(x) = \ln(1+x),$ $a = 0$     $L(x) =$    
L6. $f(x) = x^{1.3},$ $a= 1$     $L(x) =$    
L7. $f(x) = \frac{1}{1+e^{0.2x}},$ $a = 0$ $L(x) =$    
L8. $f(x) = \cos(x),$ $a = 0$ $L(x) =$    
L9. $f(x) = \sin(x),$ $a = 0$ $L(x) =$    

Use linear approximation to estimate the given numbers. (The exact values will not be accepted). You may round answers to $4$ decimal places.
A1. $\sqrt{16.3}$ $\approx$    
A2. $\sqrt{48.69}$ $\approx$    
A3. $(3.9)^{3/2}$ $\approx$    
A4. $e^{0.3}$ $\approx$    
A5. $\ln(0.95)$ $\approx$    
A6. $\sin(0.131)$ $\approx$    

App1. Quality Control Silicon Valley, Inc. manufactures blank compact discs for sale to recording studios. Its CD's have a radius of $5cm.$ A disc whose radius is off by more than $0.05cm$ is automatically rejected.
The volume of the discs that pass inspection can vary by $cm^2.$

App2. Sales The demand equation for your new fraternity T-shirts is given by where $q$ represents the weekly sales of T-shirts at a price of $p.$ You are currently charging $5$ per T-shirt.
If you raise the price to $5.05,$ your sales will drop by about shirts per week.

App3. Cost Analysis The daily cost of manufacturing $x$ camcorders at Consumer Electronics, Inc. is calculated to be
The linear approximation to $C(x)$ near $x = 100$ is $L(x) =$
The approximate cost of manufacturing $102$ camcorders is $\$$

App4. Measurement The radius of the earth is approximately $6\,400 km$ (roughly $4\,000$ miles). Suppose a cable was laid all the way around the equator on the surface of the earth.

(a) How much longer would the cable have to be if it was to be raised $1$ meter above the surface all the way around the earth?
 

(b) Repeat part (a) for a cable around Jupiter, whose approximate radius is $71\,400 km$ (roughly $44\,400$ miles).
 


 

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Last Updated:March, 2000
Copyright © 2000 StefanWaner and Steven R. Costenoble