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Introduction

Example 1

Example 1 – Continued

Example 1 – Continued

Solve \(2^{x} + x^{2} - 3 = 0\), to two decimal places of accuracy.

Solution

Using a systematic guess and check method, we can estimate the location of the other point of intersection to two decimal places of accuracy. In the interval \(-2 \lt x \lt -1\), we have

a graph displaying the function f(x) = 2^x and g(x) = -x^2 + 3

 

\(x\) \(f(x)=2^{x}\) \(g(x) = -x^{2} + 3\) \(f(x) - g(x)\)
\(-1.7\) \(2^{-1.7} \approx 0.3078\) \(-(-1.7)^{2} + 3 = 0.11\) \(0.1978\)
\(-1.6\) \(2^{-1.6} \approx 0.3299\) \(-(-1.6)^{2} + 3 \approx 0.44\) \(-0.1101\)
\(-1.63\) \(2^{-1.63} \approx 0.3231\) \(-(-1.63)^{2}+3 \approx 0.3431\) \(-0.0200\)
\(-1.64\) \(2^{-1.64} \approx 0.3209\) \(-(-1.64)^{2} + 3 = 0.3104\) \(0.0104\)

 

 

Check Your Understanding A

Use the systematic guess and check method to solve (((((f)*(_))*(d))*(i))*(s))*(p)  = (((((g)*(_))*(d))*(i))*(s))*(p).

The graph of f x = ((((((f)*(g))*(_))*(d))*(i))*(s))*(p) has been provided to assist you.

(((p)*(l))*(o))*(t)

f x = ((((((f)*(g))*(_))*(d))*(i))*(s))*(p)

 

Express answers to one decimal place of accuracy. (Separate multiple answers using a comma.)

x ≈ There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

null

Preview  

Note that (((((f)*(_))*(d))*(i))*(s))*(p) = (((((g)*(_))*(d))*(i))*(s))*(p) when ((((((f)*(g))*(_))*(d))*(i))*(s))*(p) = \(0\).

From the graph, we can see that \( $int_x1_search_lb(details...) \lt x \lt $int_x1_search_ub(details...) \). Choosing some points within this interval,

 

\[ \begin{align*} f($int_x1_search_l(details...)) &$eq1(details...) $int_y1_search_l(details...) \\ f($int_x1_search(details...)) &$eq2(details...) $int_y1_search(details...) \\ f($int_x1_search_u(details...))&$eq3(details...) $int_y1_search_u(details...) \end{align*} \]

 

Since \(f(x) \) changed sign between \( $int_x1_search_l(details...) \) and \( $int_x1_search_u(details...) \), and \( f($int_x1_search(details...)) \) $equal_close1(details...) \(0\), then one (((((((((((a)*(p))*(p))*(r))*(o))*(x))*(i))*(m))*(a))*(t))*2.718281828459045)*1.0 solution is \(x_1 = $int_x1_search(details...) \).

 

From the graph, we can see that \( $int_x2_search_lb(details...) \lt x \lt $int_x2_search_ub(details...) \). Choosing some points within this interval,

\[\begin{align*} f($int_x2_search_l(details...)) &$eq4(details...) $int_y2_search_l(details...) \\ f($int_x2_search(details...)) &$eq5(details...) $int_y2_search(details...) \\ f($int_x2_search_u(details...)) &$eq6(details...) $int_y2_search_u(details...) \end{align*}\]

 

Since \( f(x) \) changed sign between \( $int_x2_search_l(details...) \) and \( $int_x2_search_u(details...) \), and \( f($int_x2_search(details...)) \) $equal_close2(details...) \(0\), then the other (((((((((((a)*(p))*(p))*(r))*(o))*(x))*(i))*(m))*(a))*(t))*2.718281828459045)*2.0 solution is \(x_2 = $int_x2_search(details...) \).
 

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