Slope and equation of tangent line - Derivatives

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

Notes:

• Point-Slope Form of a line with slope m through a point $(x_1,y_1): m=\frac{y-y_1}{x-x_1}$

• 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.

Intro Lesson

Connecting: Derivative & Slope & Equation of Tangent Line
Exercise: The graph of the quadratic function $f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1$ is shown below.
1.
Determining Equations of the Tangent Line and Normal Line
Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$
2.
Locating Horizontal Tangent Lines
4.
Determining Lines Passing Through a Point and Tangent to a Function
Consider the quadratic function: $f(x)=x^2-x-2$
5.
Locating Lines Simultaneously Tangent to 2 Curves
Consider the quadratic functions:
$f(x)=x^2$
$g(x)=\frac{1}{4}x^2+3$

Slope and equation of tangent line

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Lessons

Notes:

Intro Lesson

Connecting: Derivative & Slope & Equation of Tangent Line Exercise: The graph of the quadratic function f(x)=12x2+2x−1f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1f(x)=​2​​1​​x​2​​+2x−1 is shown below.

a)

Find and interpret f′(x)f'\left( x \right)f​′​​(x).

b)

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

c)

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

1.

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})f(x)=​32​​x​​(√​x​​​+​3​​√​x​​​)

a)

Determine an equation of the tangent line to the curve at x=64x=64x=64.

b)

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

2.

Locating Horizontal Tangent Lines

a)

Find the points on the graph of f(x)=2x3−3x2−12x+8f(x)=2x^3-3x^2-12x+8f(x)=2x​3​​−3x​2​​−12x+8 where the tangent is horizontal.

b)

Find the vertex of each quadratic function: f(x)=2x2−12x+10f(x)=2x^2-12x+10f(x)=2x​2​​−12x+10 g(x)=−3x2−60x−50g(x)=-3x^2-60x-50g(x)=−3x​2​​−60x−50

3.

4.

Determining Lines Passing Through a Point and Tangent to a Function Consider the quadratic function: f(x)=x2−x−2f(x)=x^2-x-2f(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.

5.

Locating Lines Simultaneously Tangent to 2 Curves Consider the quadratic functions: f(x)=x2f(x)=x^2f(x)=x​2​​ g(x)=14x2+3g(x)=\frac{1}{4}x^2+3g(x)=​4​​1​​x​2​​+3

a)

Sketch each parabola.

b)

Determine the lines that are tangent to both curves.

Slope and equation of tangent line

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We have over 350 practice questions in Calculus for you to master.

or

Connecting: Derivative & Slope & Equation of Tangent Line
Exercise: The graph of the quadratic function $f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1$ is shown below.

Find and interpret $f'\left( x \right)$ .

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

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

Determining Equations of the Tangent Line and Normal Line
Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$

Determine an equation of the tangent line to the curve at $x=64$ .

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

Find the points on the graph of $f(x)=2x^3-3x^2-12x+8$ where the tangent is horizontal.

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

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

Locating Lines Simultaneously Tangent to 2 Curves
Consider the quadratic functions:
$f(x)=x^2$
$g(x)=\frac{1}{4}x^2+3$