(a) Find the tangent line approximations to the graph of f(x)= sqrt(x) near x=4

Question

(a) Find the tangent line approximations to the graph of f(x)= sqrt(x) near x=4 and x=9. Use each of them to estimate sqrt(6.5)

(b) Are the two estimates of sqrt(6.5) you found in part (a) over-or under-estimates of the exact value of sqrt(6.5)? Explain how you know this.

(c) Which of the two estimates you found in part (a) is closer to the exact value of sqrt(6.5)? Use a carefully drawn sketch of the graph of f(x)= sqrt(x) and the two tangent lines to explain how you know this.

(d) Decide where in between x=4and x=9 you should switch tangent lines to approximate square roots. That is, give an interval of x values where the tangent line approximation near x=4 will most accurately estimate sqrt(x) and another interval where the tangent line approximation near x=9 will most accurately estimate sqrt(x). You may want to refer to the graph from part (c) to help you answer this.

(e) Where should you switch tangent lines between 9 and 16? 16 and 25? Come up with a conjecture about where you should switch between any two perfect squares.

Explanation / Answer

f(x) = rtx

Hence f'(x) = 1/(2 rt x)

f'(4) = 1/2 (2) = 0.25

f'(9) =1/6

f'(4) = f(6.5) -f(4)/2.5

or f(6.5) = f(4) +2.5 (1/4)

= 2.625

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f'(9) = f(6.5)-f(9)/-2.5 = 1.5

f(6.5) = 3- 0.4167=2.833

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Both are over estimates

2.625 i.e. using 4 is closer to the real value.

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