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Tangent lines

Exercises · Derivatives · with worked solutions · PDF

Exercise 1

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = 3\:x^2 + 2\:x +1, ~~x_0=2 \]
Solution

We compute the equation of the tangent line to the graph of the function

\[ f(x) = 3\:x^2 + 2\:x +1 {\rm ~~at~the~point~with~abscissa~~} x_0 = 2, {\rm ~~we~have:} \]
\[ f(2)=17,~ f'(x)=6\:x+2,~ f'(2)=14; \]

the tangent line is

\[ y = f(2) + f'(2)(x - 2) = 17 + 14\: (x - 2) \]

Figure 1

Exercise 2

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = \sin x, ~~x_0=\frac{\pi}{3} \]
Solution

We compute the equation of the tangent line to the graph of the function

\[ f(x) = \sin x {\rm ~~at~the~point~with~abscissa~~} x_0 = \frac{\pi}{3}, {\rm ~~we~have:} \]
\[ f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2},~ f'(x)=\cos x,~ f'\left(\frac{\pi}{3}\right)=\frac{1}{2}; \]

the tangent line is

\[ y = f\left(\frac{\pi}{3}\right) + f'\left(\frac{\pi}{3}\right)\left(x - \frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} + \frac{1}{2}\: \left(x - \frac{\pi}{3}\right) \]

Figure 2

Exercise 3

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = \log x, ~~x_0=1 \]
Solution

We compute the equation of the tangent line to the graph of the function

\[ f(x) = \log x {\rm ~~at~the~point~with~abscissa~~} x_0 = 1, {\rm ~~we~have:} \]
\[ f(1)=0,~ f'(x_0)=\lim_{h \rr 0} \frac{f(x_0+h)-f(x_0)}{h} =\lim_{h \rr 0}\frac{\log (1+h)}{h} = 1 \]

the tangent line is

\[ y = f(1) + f'(1)(x - 1) = 0 + (x - 1) \]

Figure 3

Exercise 4

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = a^x, ~~x_0=2 \]

Exercise 5

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = (x \: \log |x|)^3, ~~x_0=-1 \]

Exercise 6

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = \cos (\log x), ~~x_0=e^{\frac{\pi}{2}} \]

Exercise 7

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = e^{x^2}, ~~x_0=\log 2 \]

Exercise 8

Write the equation of the tangent line to the graph of \(y = f(x)\) at the point \(\big( x_0, f(x_0)\big)\) for

\[ f(x) = e^{-|x|}, ~~x_0=-1 \]