Tangents and Normals PDF

TANGENTS AND NORMALS EQUATION OF THE TANGENT LINE The derivative of a function at point 𝑥 = 𝑎, denoted 𝑓′(𝑎), is the instantaneous rate of change at that point. Geometrically, 𝑓 ′ (𝑎) is the slope of the tangent line at the point where 𝑥 = 𝑎. To find the equation of the tangent line, we need its s...

TANGENTS AND NORMALS EQUATION OF THE TANGENT LINE The derivative of a function at point 𝑥 = 𝑎, denoted 𝑓′(𝑎), is the instantaneous rate of change at that point. Geometrically, 𝑓 ′ (𝑎) is the slope of the tangent line at the point where 𝑥 = 𝑎. To find the equation of the tangent line, we need its slope and a point on the line. We can then use the point-slope form of the equation of the line 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏 ) where 𝑚 = 𝑓′(𝑎) at point 𝑥 = 𝑥1. STEPS 1. Find the point of tangency. 2. Find the slope of the tangent line; that is, find the derivative of the function at the the point of tangency (𝑥 = 𝑥1 ). 3. Using the slope and the point, use the point-slope form to solve for the equation of the tangent line. TANGENTS AND NORMALS Find an equation of the tangent line to the graph of the function 𝑓 𝑥 = 𝑥 2 at 𝑥 = 3. TANGENTS AND NORMALS Find the tangent line to the curve 𝑅 𝑧 = 5𝑧 − 8 at 𝑧 = 3. TANGENTS AND NORMALS Find an equation of the tangent line to the graph of the function 𝑓 𝑥 = 𝑥𝑒 𝑥 at 𝑥 = 0. TANGENTS AND NORMALS One form for the equation of a straight line is 𝒚 = 𝒎𝒙 + 𝒄 where 𝒄 is the value where the line intersects the 𝑦 −axis and 𝒎 is the gradient (slope) of the line, and 𝒎 = 𝐭𝐚𝐧 𝜽 If we have a second line 𝑦 = 𝑛𝑥 + 𝑑 perpendicular to the first line, then the slopes of the lines have the relationship 𝟏 𝒎𝟏 = − 𝒎𝟐 𝟏 𝒎𝟐 = − 𝒎𝟏 TANGENTS AND NORMALS Find the equation of the normal to the curve 𝟏 𝒚=𝒙+ 𝒙 At the point where 𝑥 = 2. TANGENTS AND NORMALS Find the equation of the normal to the curve 𝒇 𝒙 = 𝟐𝒙𝟑 − 𝟓𝐱 + 𝟒 At the point (−1, 7). TANGENTS AND NORMALS Find the equation of the tangent to the curve 𝑥 2 + 2𝑦 = 8 which is perpendicular to the line 𝑥 − 2𝑦 + 1 = 0. 𝑦 = −2𝑥 − 2