Given a function, a {{CMD_7}}linear approximation{{CMD_8}} is a fancy phrase for something you already know: {{CMD_9}} {{CMD_10}}The line tangent to the graph of a function at a point is very close to the graph of the function near that point.{{CMD_11}} {{CMD_12}} This tangent line is the graph of a linear function, called the {{CMD_13}}linear approximation{{CMD_14}}. {{CMD_15}} Let {{MATH_16}} be a function that is differentiable on some interval I that contains the point {{MATH_17}}. The graph of a function {{MATH_18}} and the line tangent to the curve {{MATH_19}} at the point where {{MATH_20}} are given in the figure below. Find the equation of the tangent line. {{CMD_21}} {{ENV_22}} {{COMMENT_23}} {{COMMENT_24}} {{CMD_25}} First, find the expression for {{MATH_26}}, the slope of the tangent line to the curve {{MATH_27}} at the point {{MATH_28}}. Select the correct choice. {{CMD_29}} {{CMD_30}}{{MATH_31}}{{CMD_32}} {{CMD_33}}{{MATH_34}}{{CMD_35}} {{CMD_36}}{{MATH_37}}{{CMD_38}} {{CMD_39}}{{MATH_40}}{{CMD_41}} {{CMD_42}} We don't have enough information to determine the slope.{{CMD_43}} {{CMD_44}} Since we know that the point {{MATH_45}} lies on the tangent line, we can write an equation of the tangent line. {{MATH_46}} Now, we define a function, {{MATH_47}}, by {{MATH_48}}. This function is linear and its graph is the line tangent to the curve {{MATH_49}} at the point where {{MATH_50}}. This function deserves a special name. {{CMD_51}} {{CMD_52}}{{CMD_53}} If {{MATH_54}} is a function differentiable at {{MATH_55}}, then a {{CMD_56}}linear approximation{{CMD_57}} to the function {{MATH_58}} at {{MATH_59}} is given by {{MATH_60}} {{CMD_61}}