The Point $P ( 1 , \sqrt { 3 ) }$

question 28

Essay

The point $P ( 1 , \sqrt { 3 ) }$ lies on the curve $y = \sqrt { 4 - x ^ { 2 } }$ Let Q be the point $\left( x , \sqrt { 4 - x ^ { 2 } } \right)$ (a) What is the slope of the secant line PQ (correct to 6 decimal places) for the following values of x?
(i) 2
(ii) 1.5
(iii) 1.1
(iv) 1.01
(v) 1.001
(vi) 0
(vii) 0.5
(viii) 0.9
(ix) 0.99
(x) 0.999
(b) Use your results from part (a) to estimate the slope of the tangent line to the graph of $y = \sqrt { 4 - x ^ { 2 } \text { at } x = 1 }$

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Definitions:

Conscious Thought

The state of being aware of one's own existence, sensations, thoughts, surroundings, and the ability to actively process this information.

Classical Conditioning

A learning process that occurs when two stimuli are repeatedly paired: a response that is at first elicited by the second stimulus is eventually elicited by the first stimulus alone.

Behaviorism

Behaviorism is a theory of learning which states that all behaviors are acquired through conditioning, emphasizing the role of external stimuli and responses without considering thoughts or feelings.

Mental Explanations

Interpretations or reasons provided to understand behaviors or phenomena, focusing on psychological, rather than physical or external, factors.