Horizontal Tangent Line

A horizontal tangent line occurs when the slope of a function is zero at a point on the graph. To find horizontal tangent lines, take the derivative, set it equal to zero, solve for the x-values, and then plug those x-values back into the original function. The equations of horizontal tangent lines are always in the form \(y=c\).

Notes

Calculus Horizontal Tangent Line

Practice Problems

Find the equations of the horizontal tangent lines.

\(\textbf{1)}\) \( f(x)=x^2+4x+4 \)Link to Youtube Video Solving Question Number 1

 

\(\textbf{2)}\) \( f(x)=\sin ⁡x \)Link to Youtube Video Solving Question Number 2

 

\(\textbf{3)}\) \( f(x)=4 \)Link to Youtube Video Solving Question Number 3

 

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

 

\(\textbf{5)}\) \( f(x) = x^2 – 4 \)

 

\(\textbf{6)}\) \( f(x) = x^3 – 3x \)

 

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

 

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

 

\(\textbf{9)}\) \( f(x)=x^3-6x^2+9x \)

 

\(\textbf{10)}\) \( f(x)=x^4-8x^2 \)

 

\(\textbf{11)}\) \( f(x)=e^x \)

 

\(\textbf{12)}\) \( f(x)=\ln{x} \)

 

\(\textbf{13)}\) \( f(x)=x+\frac{4}{x} \)

 

\(\textbf{14)}\) \( f(x)=\sqrt{x+4} \)

 

\(\textbf{15)}\) \( f(x)=\frac{x^2}{x+1} \)

 

\(\textbf{16)}\) If a function has a local minimum, the derivative at that point is positive.

 

\(\textbf{17)}\) The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.

 

\(\textbf{18)}\) If the derivative of a function is zero at a point, the tangent line to the graph at that point is horizontal.

 

\(\textbf{19)}\) A function with a positive derivative at a point must have a local maximum at that point.

 

\(\textbf{20)}\) The tangent line to a constant function graph is horizontal at every point.

 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graph of First and Second Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

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