Applications of Derivatives: Training for IITJEE Maths: Part V

Question I:

Show that the equation of the tangent to the curve $x=a \frac{f(t)}{h(t)}$ and $y=a \frac{g(t)}{h(t)}$ can be represented in the form: $\left| \begin{array}{ccc} x & y & a \\ f(t) & g(t) & h(t) \\ f^{'}(t) & g^{'}(t) & h^{'}(t) \end{array} \right|=0$

Question 2:

Show that the derivative of the function $f(x) = x \sin{\frac{\pi}{x}}$, when $x>0$ and $f(x)=0$, when $x=0$ vanishes on an infinite set of points of the interval \$latex (0,1), Hint: Use Rolle’s theorem.

Question 3:

Prove that $\frac{1}{1+x}<\log (1+x) < x$ for $x>0$. Use Lagrange’s theorem.

Question 4:

Find the largest term in the sequence $a_{n}=\frac{n^{2}}{n^{3}+200}$. Hint: Consider the function $f(x)=\frac{x^{2}}{x^{3}+200}$ in the interval $[1,\infty)$.

Question 5:

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Question 6:

Find the polynomial $f(x)$ of degree 6, which satisfies $\lim_{x \rightarrow 0} (1+\frac{f(x)}{x^{3}})=e^{2}$ and has a local maximum at $x=1$ and local minimum at $x=0$ and 2.

Question 7:

For the circle $x^{2}+y^{2}=r^{2}$, find the value of r for which the area enclosed by the tangents drawn from the point $(6,8)$ to the circle and the chord of contact is maximum.

Question 8:

Suppose that f has a continuous derivative for all values of x and $f(0)=0$, with $|f^{'}(x)| <1$ for all x. Prove that $|f(x)| \leq |x|$.

Question 9:

Show that $(e^{x}-1)>(1+x)\log {1+x}$, if $x \in (0,\infty)$.

Question 10:

Let $-1 \leq p \leq 1$. Show that the equations $4x^{3}-3x-p=0$ has a unique root in the interval $[1/2,1]$ and identify it.

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