Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet (Page 23 of 41) in pdf

Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 23

Worksheet # 17: Linear Approximation and Applications

1. For each of the following, use a linear approximation to the change in the function and a convenient

nearby point to estimate the value:

(a) (3.01)

(b)

(d) tan(44 )

2. What is the relation between the linearization of a function f (x) at x = a and the tangent line to the

graph of the function f (x) at x = a on the graph?

3. Use the linearization of

x at x = 16 to estimate

18;

(a) Find a decimal approximation to

18 using a calculator.

(b) Compute both the absolute error and the percentage error if we use the linearization to approxi-

mate

18.

4. Suppose we want to paint a sphere of radius 200 cm with a coat of paint 0.1 mm thick. Use a linear

approximation to approximate the amount of paint we need to do the job.

5. Let f (x) =

16 + x. First, ﬁnd the linearization to f (x) at x = 0, then use the linearization to

estimate

15.75. Present your solution as a rational number.

6. Find the linearization L(x) to the function f (x) =

2x at x =

7. Find the linearization L(x) to the function f (x) =

x + 4 at x = 4, then use the linearization to

estimate

8.25.

8. Your physics professor tells you that you can replace sin(θ) with θ when θ is close to zero. Explain

why this is reasonable.

9. Suppose we measure the radius of a sphere as 10 cm with an accuracy of

0.2 cm. Use linear

approximations to estimate the maximum error in:

(a) the computed surface area.

(b) the computed volume.

10. Suppose that y = y(x) is a diﬀerentiable function which is deﬁned near x = 2, satisﬁes y(2) =

1 and

x + 3xy + y = 9.

Use the linear approximation to the change in y to approximate the value of y(1.91).

11. (Review) Can you ﬁnd a function f so that f (x) = 2x + 1? Can you ﬁnd a function g so that

g (x) = sin(x) + cos(x)? Can you ﬁnd more than one answer?

12. (Review) Find the derivative of sin(sin(sin(sin(x)))).

Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 23