Application of Derivatives

The slope of the tangent to the curve x = t² + 3t - 8, y = 2t² - 2t - 5 at the point (2, - 1) is

• $\frac{22}{7}$

• $\frac{6}{7}$

• $\frac{7}{6}$

• $\frac{- 6}{7}$

The equation of the normal to the curve y(1 + x²) = 2 - x, where the tangent crosses X-axis, is

• 5x - y - 10 = 0

• x - 5y - 10 = 0

• 5x + y + 10 = 0

• x + 5y + 10 = 0

The maximum value of ${\left(\frac{1}{\mathrm{x}}\right)}^{\mathrm{x}}$ is

• e

• e^e

• e^1/e

• ${\left(\frac{1}{\mathrm{e}}\right)}^{\mathrm{e}}$

The rate of change of area of a circle with  respect to its radius at r = 2 cm is

• 4

• 2$\mathrm{\pi }$

• 2

• 4$\mathrm{\pi }$

The function f(x) = x² + 2x - 5 is strictly increasing in the interval

• $(- \infty , - 1)$

• $[- 1, \infty )$

• $(- \infty , 1]$

• $(- 1, \infty )$

The point on the curve y² = x where the tangent makes an angle of $\mathrm{\pi }$/4 with X-axis is

• (4, 2)

• $\left(\frac{1}{2}, \frac{1}{4}\right)$

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• (1, 1)

The rate of change ofvolume of a sphere with respect to its surface area when the radius is 4 cm is

• 4 cm³/cm²

• 6 cm³/cm²

• 2 cm³/cm²

• 8 cm³/cm²

The two curves x³ - 3xy² + 2 = 0 and 3x²y - y³ - 2 = 0

• cut at right angle

• touch each other

• cut at an angle $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• cut at an angle $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

The function f(x) = cot^-1(x) + x, increases in the interval

• $\left(1, \infty \right)$

• $\left(- 1, \infty \right)$

• $\left(- \infty , \infty \right)$

• $\left(0, \infty \right)$

For all real values of x, increasing function is

• x^{- 1}

• x²

• x³

• x⁴