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Questions

\(\textbf{1)}\) Find the equation of the tangent line to the curve \( f(x)=x^3+3x^2-x \) at the point \( (2,18) \)

Show Answer

The answer is \( y=23x-28 \)

Show Work

\(\,\,\,\text{Step 1: Find the slope with the derivative.}\)

\(\,\,\,\,\,\,f(x)=x^3+3x^2-x\)

\(\,\,\,\,\,\,f'(x)=3x^2+6x-1\)

\(\,\,\,\,\,\,f'(2)=3(2)^2+6(2)-1\)

\(\,\,\,\,\,\,f'(2)=12+12-1\)

\(\,\,\,\,\,\,f'(2)=23\)

\(\,\,\,\,\,\,\text{Slope at } (2,18) \text{ is }m=23\)

\(\,\,\,\text{Step 2: Use the point and slope to find the equation of the line.}\)

\(\,\,\,\,\,\,\text{Point Slope Formula: }y-y_1=m(x-x_1)\)

\(\,\,\,\,\,\,y-18=23(x-2)\)

\(\,\,\,\,\,\,y-18=23x-46\)

\(\,\,\,\,\,\,\text{The answer is } y=23x-28 \)

\(\textbf{2)}\) Find the equation of the tangent line to the curve \( f(x)=x\sqrt{x} \) at the point \((4,8) \)

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The answer is \( y=3x-4 \)

Show Work

\(\,\,\,\text{Step 1: Find the slope with the derivative.}\)

\(\,\,\,\,\,\,\,\,\, f(x)=x\sqrt{x} \)

\(\,\,\,\,\,\,\,\,\, f(x)=x^{3/2} \)

\(\,\,\,\,\,\,\,\,\, f'(x)=\frac{3}{2}x^{1/2} \)

\(\,\,\,\,\,\,\,\,\, f'(4)=\frac{3}{2}(4)^{1/2} \)

\(\,\,\,\,\,\,\,\,\, f'(4)=\frac{3}{2}(2) \)

\(\,\,\,\,\,\,\,\,\, f'(4)=3 \)

\(\,\,\,\,\,\,\,\,\,\text{Slope at } (4,8) \text{ is }m=3\)

\(\,\,\,\text{Step 2: Use the point and slope to find the equation of the line.}\)

\(\,\,\,\,\,\,\,\,\,\text{Point Slope Formula: }y-y_1=m(x-x_1)\)

\(\,\,\,\,\,\,\,\,\,y-8=3(x-4)\)

\(\,\,\,\,\,\,\,\,\,y-8=3x-12\)

\(\,\,\,\,\,\,\,\,\,y=3x-4\)

\(\,\,\,\)The answer is \( y=3x-4 \)

\(\textbf{3)}\) Find the equation of the tangent line to the curve \( f(x)=x^2-6x+4 \) and parallel to the line \(y=-4x+9 \)

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The answer is \( y=-4x+3 \)

\(\textbf{4)}\) Find the equation of the tangent line to the curve \( f(x)=\frac{e^x}{x} \) at the point \( (1,e) \)

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The answer is \( y=e \)

\(\textbf{5)}\) Find the equation of the tangent line to the curve \( f(x)=5e^x \cos⁡{x} \) at the point \( (0,5) \)

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The answer is \( y=5x+5 \)

\(\textbf{6)}\) Find the equation of the tangent line to the curve \( f(x)=5\sqrt{x}(x+4) \) at the point \( (4,80) \)

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The answer is \( y=20x\)

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

Tangent lines are a key concept in calculus. The slope of a tangent line is same as the instantaneous slope (or derivative) of the graph at that point. We can find the equation of the tangent line by using point slope formula \(y-y_0=m\left(x-x_0\right)\), where we use the derivative value for the slope and the point of tangency as the point \(\left(x_0,y_0\right)\).

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