Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Notes

Questions

Find the equations of the horizontal tangent lines.

\(\textbf{1)}\) \( f(x)=x^2+4x+4 \)

Show Work

\(\,\,\,\,\,f'(x)=2x+4\)
\(\,\,\,\,\,2x+4=0\)
\(\,\,\,\,\,2x=-4\)
\(\,\,\,\,\,x=-2\)

\(\,\,\,\,\,f(-2)=(-2)^2+4(-2)+4\)
\(\,\,\,\,\,f(-2)=4-8+4\)
\(\,\,\,\,\,f(-2)=0\)

Show Answer

The answer is \( y=0 \)

\(\textbf{2)}\) \( f(x)=\sin ⁡x \)

Show Answer

The answer is \( y=1 \) and \( y=-1 \)

\(\textbf{3)}\) \( f(x)=4 \)

Show Answer

The answer is \( y=4 \)

\(\textbf{4)}\) \( f(x) = x^3 – 3x^2 + 3x \)

Show Work

To find where the tangent line is horizontal, we first take the derivative:
\(\,\,\,\,\,f'(x) = 3x^2 – 6x + 3\)

Set the derivative equal to 0 to find critical points:
\(\,\,\,\,\,3x^2 – 6x + 3 = 0\)

Factor the quadratic:
\(\,\,\,\,\,3(x^2 – 2x + 1) = 0\)
\(\,\,\,\,\,(x – 1)^2 = 0\)
\(\,\,\,\,\,x = 1\)

Now find the corresponding \( y \) value:
\(\,\,\,\,\,f(1) = (1)^3 – 3(1)^2 + 3(1) = 1 – 3 + 3 = 1\)

Show Answer

The answer is \( y=1 \)

\(\textbf{5)}\) \( f(x) = x^2 – 4 \)

Show Work

To find where the tangent line is horizontal, we first take the derivative:
\(\,\,\,\,\,f'(x) = 2x\)

Set the derivative equal to 0:
\(\,\,\,\,\,2x = 0\) \(\,\,\,\,\,x = 0\)

Now find the corresponding \( y \) value:
\(\,\,\,\,\,f(0) = (0)^2 – 4 = -4\)

Show Answer

The answer is \( y=-4 \)

\(\textbf{6)}\) \( f(x) = x^3 – 3x \)

Show Work

To find where the tangent line is horizontal, we first take the derivative:
\(\,\,\,\,\,f'(x) = 3x^2 – 3\)

Set the derivative equal to 0:
\(\,\,\,\,\,3x^2 – 3 = 0\)
\(\,\,\,\,\,3(x^2 – 1) = 0\)
\(\,\,\,\,\,x^2 = 1\)
\(\,\,\,\,\,x = \pm 1\)

Now find the corresponding \( y \) values:
\(\,\,\,\,\,f(1) = (1)^3 – 3(1) = 1 – 3 = -2\)
\(\,\,\,\,\,f(-1) = (-1)^3 – 3(-1) = -1 + 3 = 2\)

Show Answer

The answers are \( y = -2 \) and \( y = 2 \)

\(\textbf{7)}\) \( f(x) = \cos x \)

Show Work

To find the horizontal tangent lines, first take the derivative:
\(\,\,\,\,\,f'(x) = -\sin x\)

Set the derivative equal to 0:
\(\,\,\,\,\, -\sin x = 0 \)
\(\,\,\,\,\, \sin x = 0 \)

The sine function is 0 at \( x = 0, \pi, 2\pi, \dots \)

Now find the corresponding \( y \) values:
\(\,\,\,\,\,f(0) = \cos(0) = 1\)
\(\,\,\,\,\,f(\pi) = \cos(\pi) = -1\)

Show Answer

The answers are \( y=1 \) and \( y=-1 \)

\(\textbf{8)}\) \( f(x) = 2x^2 – 8x + 6 \)

Show Work

To find where the tangent line is horizontal, first take the derivative:
\(\,\,\,\,\,f'(x) = 4x – 8\)
Set the derivative equal to 0:
\(\,\,\,\,\,4x – 8 = 0\)
\(\,\,\,\,\,4x = 8\)
\(\,\,\,\,\,x = 2\)
Now find the corresponding \( y \) value:
\(\,\,\,\,\,f(2) = 2(2)^2 – 8(2) + 6 = 8 – 16 + 6 = -2\)

Show Answer

The answer is \( y=-2 \)

True or False

\(\textbf{9)}\) If a function has a local minimum, the derivative at that point is positive.

True or False

The statement is False.

\(\textbf{10)}\) The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.

True or False

The statement is True.

\(\textbf{11)}\) If the derivative of a function is zero at a point, the tangent line to the graph at that point is horizontal.

True or False

The statement is True.

\(\textbf{12)}\) A function with a positive derivative at a point must have a local maximum at that point.

True or False

The statement is False.

\(\textbf{13)}\) The tangent line to a constant function graph is horizontal at every point.

True or False

The statement is True.

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

In Summary

A horizontal tangent line occurs at points where the instantaneous slope of a function is zero. We can find it by taking the derivative of a function, setting it equal to zero, and solving for x. This topic is usually studied in calculus courses along with derivatives.

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!