Question

suppose that f(500) = 8000 and f’(500) = 30. estimate each of the following. (a) f(501) (b) f(500.5) (c) f(499) (d) f(498) (e) f(499.75)

Explanation:

To solve this problem, we use the linear approximation formula, which is based on the idea that for a function \( f(x) \) that is differentiable at \( x = a \), the value of \( f(x) \) near \( a \) can be approximated by:

$$ f(x) \approx f(a) + f'(a)(x - a) $$

where \( a = 500 \), \( f(a) = 8000 \), and \( f'(a) = 30 \). We will apply this formula to each part (a) to (e).

Part (a): Estimate \( f(501) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 501 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(501) \approx f(500) + f'(500)(501 - 500) $$

$$ f(501) \approx 8000 + 30(1) $$

$$ f(501) \approx 8000 + 30 $$

$$ f(501) \approx 8030 $$

Part (b): Estimate \( f(500.5) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 500.5 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(500.5) \approx f(500) + f'(500)(500.5 - 500) $$

$$ f(500.5) \approx 8000 + 30(0.5) $$

$$ f(500.5) \approx 8000 + 15 $$

$$ f(500.5) \approx 8015 $$

Part (c): Estimate \( f(499) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 499 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(499) \approx f(500) + f'(500)(499 - 500) $$

$$ f(499) \approx 8000 + 30(-1) $$

$$ f(499) \approx 8000 - 30 $$

$$ f(499) \approx 7970 $$

Part (d): Estimate \( f(498) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 498 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(498) \approx f(500) + f'(500)(498 - 500) $$

$$ f(498) \approx 8000 + 30(-2) $$

$$ f(498) \approx 8000 - 60 $$

$$ f(498) \approx 7940 $$

Part (e): Estimate \( f(499.75) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 499.75 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(499.75) \approx f(500) + f'(500)(499.75 - 500) $$

$$ f(499.75) \approx 8000 + 30(-0.25) $$

$$ f(499.75) \approx 8000 - 7.5 $$

$$ f(499.75) \approx 7992.5 $$

Final Answers

(a) \( \boldsymbol{f(501) \approx 8030} \)
(b) \( \boldsymbol{f(500.5) \approx 8015} \)
(c) \( \boldsymbol{f(499) \approx 7970} \)
(d) \( \boldsymbol{f(498) \approx 7940} \)
(e) \( \boldsymbol{f(499.75) \approx 7992.5} \)

Answer:

To solve this problem, we use the linear approximation formula, which is based on the idea that for a function \( f(x) \) that is differentiable at \( x = a \), the value of \( f(x) \) near \( a \) can be approximated by:

$$ f(x) \approx f(a) + f'(a)(x - a) $$

where \( a = 500 \), \( f(a) = 8000 \), and \( f'(a) = 30 \). We will apply this formula to each part (a) to (e).

Part (a): Estimate \( f(501) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 501 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(501) \approx f(500) + f'(500)(501 - 500) $$

$$ f(501) \approx 8000 + 30(1) $$

$$ f(501) \approx 8000 + 30 $$

$$ f(501) \approx 8030 $$

Part (b): Estimate \( f(500.5) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 500.5 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(500.5) \approx f(500) + f'(500)(500.5 - 500) $$

$$ f(500.5) \approx 8000 + 30(0.5) $$

$$ f(500.5) \approx 8000 + 15 $$

$$ f(500.5) \approx 8015 $$

Part (c): Estimate \( f(499) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 499 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(499) \approx f(500) + f'(500)(499 - 500) $$

$$ f(499) \approx 8000 + 30(-1) $$

$$ f(499) \approx 8000 - 30 $$

$$ f(499) \approx 7970 $$

Part (d): Estimate \( f(498) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 498 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(498) \approx f(500) + f'(500)(498 - 500) $$

$$ f(498) \approx 8000 + 30(-2) $$

$$ f(498) \approx 8000 - 60 $$

$$ f(498) \approx 7940 $$

Part (e): Estimate \( f(499.75) \)

Step 1: Identify \( a \), \( x \), \( f(a) \), and \( f'(a) \)

Here, \( a = 500 \), \( x = 499.75 \), \( f(a) = 8000 \), and \( f'(a) = 30 \).

Step 2: Apply the linear approximation formula

$$ f(499.75) \approx f(500) + f'(500)(499.75 - 500) $$

$$ f(499.75) \approx 8000 + 30(-0.25) $$

$$ f(499.75) \approx 8000 - 7.5 $$

$$ f(499.75) \approx 7992.5 $$

Final Answers

(a) \( \boldsymbol{f(501) \approx 8030} \)
(b) \( \boldsymbol{f(500.5) \approx 8015} \)
(c) \( \boldsymbol{f(499) \approx 7970} \)
(d) \( \boldsymbol{f(498) \approx 7940} \)
(e) \( \boldsymbol{f(499.75) \approx 7992.5} \)