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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