Slope and equation of tangent line

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=yy1xx1(x_1,y_1): m=\frac{y-y_1}{x-x_1}

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.
  • 1.
    Connecting: Derivative & Slope & Equation of Tangent Line
    Exercise: The graph of the quadratic function f(x)=12x2+2x1f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1 is shown below.
    Slope and equation of tangent line
    a)
    Find and interpret f(x)f'\left( x \right).

    b)
    Find the slope of the tangent line at:
    i) x=1x = - 1
    ii) x=2x = 2
    iii) x=7x = - 7
    iv) x=4x = - 4
    v) x=2x = - 2

    c)
    Find an equation of the tangent line at:
    i) x=2x = 2
    ii) x=4x = - 4
    iii) x=2x = - 2


  • 2.
    Determining Equations of the Tangent Line and Normal Line
    Consider the function: f(x)=x32(x+3x)f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})
    a)
    Determine an equation of the tangent line to the curve at x=64x=64.

    b)
    Determine an equation of the normal line to the curve at x=64x=64.


  • 3.
    Locating Horizontal Tangent Lines
    a)
    Find the points on the graph of f(x)=2x33x212x+8f(x)=2x^3-3x^2-12x+8 where the tangent is horizontal.

    b)
    Find the vertex of each quadratic function:
    f(x)=2x212x+10f(x)=2x^2-12x+10
    g(x)=3x260x50g(x)=-3x^2-60x-50


  • 4.
    Locating Tangent Lines Parallel to a Linear Function
    Consider the Cubic function: f(x)=x33x2+3xf(x)=x^3-3x^2+3x
    i) Find the points on the curve where the tangent lines are parallel to the line 12xy9=012x-y-9=0.
    ii) Determine the equations of these tangent lines.

  • 5.
    Determining Lines Passing Through a Point and Tangent to a Function
    Consider the quadratic function: f(x)=x2x2f(x)=x^2-x-2
    a)
    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.


  • 6.
    Locating Lines Simultaneously Tangent to 2 Curves
    Consider the quadratic functions:
    f(x)=x2f(x)=x^2
    g(x)=14x2+3g(x)=\frac{1}{4}x^2+3
    a)
    Sketch each parabola.

    b)
    Determine the lines that are tangent to both curves.