Applications of Derivatives: Tutorial Set 1: IITJEE Mains Maths

“Easy” questions:

Question 1:

Find the slope of the tangent to the curve represented by the curve $x=t^{2}+3t-8$ and $y=2t^{2}-2t-5$ at the point $(2,-1)$ .

Question 2:

Find the co-ordinates of the point P on the curve $y^{2}=2x^{3}$ , the tangent at which is perpendicular to the line $4x-3y+2=0$ .

Question 3:

Find the co-ordinates of the point $P(x,y)$ lying in the first quadrant on the ellipse $x^{2}/8 + y^{2}/18=1$ so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest.

Question 4:

The function $f(x) = \frac{\log (\pi+x)}{\log (e+x)}$ , where $x \geq 0$ is

(a) increasing on $(-\infty, \infty)$

(b) decreasing on $[0, \infty)$

(d) decreasing on $[0, \pi/e)$ and increasing on $[\pi/e, \infty)$ .

Fill in the correct multiple choice. Only one of the choices is correct.

Question 5:

Find the length of a longest interval in which the function $3\sin(x) -4\sin^{3}(x)$ is increasing.

Question 6:

Let $f(x)=x e^{x(1-x)}$ , then $f(x)$ is

(a) increasing on $[-1/2, 1]$

(b) decreasing on $\Re$

(d) decreasing on $[-1/2, 1]$ .

Fill in the correct choice above. Only one choice holds true.

Question 7:

Consider the following statements S and R:

S: Both $\sin(x)$ and $\cos (x)$ are decreasing functions in the interval $(\pi/2, \pi)$ .

R: If a differentiable function decreases in the interval $(a,b)$ , then its derivative also decreases in $(a,b)$ .

Which of the following is true?

(i) Both S and R are wrong.

(ii) Both S and R are correct, but R is not the correct explanation for S.

(iii) S is correct and R is the correct explanation for S.

(iv) S is correct and R is wrong.

Indicate the correct choice. Only one choice is correct.

Question 8:

For which of the following functions on $[0,1]$ , the Lagrange’s Mean Value theorem is not applicable:

(i) $f(x) = 1/2 -x$ , when $x<1/2$ ; and $f(x) = (1/2-x)^{2}$ , when $x \geq 1/2$ .

(ii) $f(x) = \frac{\sin(x)}{x}$ , when $x \neq 0$ ; and $f(x)=1$ , when $x=0$ .

(iii) $f(x)=x |x|$

(iv) $f(x)=|x|$ .

Only one choice is correct. Which one?

Question 9:

How many real roots does the equation $e^{x-1}+x-2=0$ have?

Question 10:

What is the difference between the greatest and least values of the function $f(x) = \cos(x) + \frac{1}{2}\cos(2x) -\frac{1}{3}\cos(3x)$ ?

More later,

Nalin Pithwa.

Mathematics Hothouse