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- Derivatives
Slope and equation of tangent line
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- Lesson: 2a11:57
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Slope and equation of tangent line
The derivative of a function is interpreted as the slope of the tangent line to the curve of the function at a certain given point. In this section, we will explore the meaning of a derivative of a function, as well as learning how to find the slope-point form of the equation of a tangent line, as well as normal lines, to a curve at multiple given points.
Lessons
• Point-Slope Form of a line with slope m through a point (x1,y1):m=x−x1y−y1
• Tangent Line & Normal Line
The normal line to a curve at a particular point is the line through that point and perpendicular to the tangent line.
• Tangent Line & Normal Line
The normal line to a curve at a particular point is the line through that point and perpendicular to the tangent line.
- IntroductionConnecting: Derivative & Slope & Equation of Tangent Line
Exercise: The graph of the quadratic function f(x)=21x2+2x−1 is shown below.
a)Find and interpret f′(x).b)Find the slope of the tangent line at:
i) x=−1
ii) x=2
iii) x=−7
iv) x=−4
v) x=−2c)Find an equation of the tangent line at:
i) x=2
ii) x=−4
iii) x=−2
- 1.Determining Equations of the Tangent Line and Normal Line
Consider the function: f(x)=32x(x+3x)a)Determine an equation of the tangent line to the curve at x=64.b)Determine an equation of the normal line to the curve at x=64. - 2.Locating Horizontal Tangent Linesa)Find the points on the graph of f(x)=2x3−3x2−12x+8 where the tangent is horizontal.b)Find the vertex of each quadratic function:
f(x)=2x2−12x+10
g(x)=−3x2−60x−50 - 3.Locating Tangent Lines Parallel to a Linear Function
Consider the Cubic function: f(x)=x3−3x2+3x
i) Find the points on the curve where the tangent lines are parallel to the line 12x−y−9=0.
ii) Determine the equations of these tangent lines. - 4.Determining Lines Passing Through a Point and Tangent to a Function
Consider the quadratic function: f(x)=x2−x−2a)Draw two lines through the point (3, -5) that are tangent to the parabola.b)Find the points where these tangent lines intersect the parabola.c)Determine the equations of both tangent lines. - 5.Locating Lines Simultaneously Tangent to 2 Curves
Consider the quadratic functions:
f(x)=x2
g(x)=41x2+3a)Sketch each parabola.b)Determine the lines that are tangent to both curves.