application-of-derivatives

Important points

• for tangent and normal equations, there can be 3 types of questions
  • given will be a point on a curve and asked will be equation of the tangent to that curve through that point
  • given will be a line and curve, and asked will be the equation of tangent to that curve parallel to that line
  • given will be a curve and a point outside the curve and asked will be the equation of tangent to that curve through that point
• equation of tangent $y-y_1=\frac{dy}{dx}{|}_{(x_1,y_1)}\times (x-x_1)$
• equation of normal $y-y_1=\frac{-1}{\frac{dy}{dx}{|}_{(x_1,y_1)}}\times (x-x_1)$ $y-y_1=-\frac{dx}{dy}{|}_{(x_1,y_1)}\times (x-x_1)$
• equation of tangent to a parabola $y^2=4ax$ having given slope m $y=mx+\frac{a}{m}$

IMPORTANT

the point on a curve which is nearest to a given line is there the tangent to that curve is parallel to the given line

Shape	TSA Formula	CSA Formula	Volume Formula
Line	N/A	N/A	N/A
Square	$4a$	N/A	N/A
Rectangle	$2(l+b)$	N/A	N/A
Circle	N/A	$2\pi r$	N/A
Triangle	$a+b+c$	N/A	N/A
Cube	$6a^2$	N/A	$a^3$
Cuboid	$2(lb+bh+hl)$	N/A	$l\times b\times h$
Sphere	$4\pi r^2$	$4\pi r^2$	$\frac{4}{3}\pi r^3$
Cylinder	$2\pi r(r+h)$	$2\pi rh$	$\pi r^{2}h$
Cone	$\pi r(r+l)$	$\pi rl$	$\frac{1}{3}\pi r^{2}h$
Hemisphere	$3\pi r^2$	$2\pi r^2$	$\frac{2}{3}\pi r^3$
Rectangular Prism	$2(lw+lh+wh)$	N/A	$l\times w\times h$
Pyramid	$\text{Base Area}+\text{CSA}$	$\frac{1}{2}\times \text{Perimeter}\times l$	$\frac{1}{3}\times \text{Base Area}\times h$
Torus	$4{\pi }^{2}Rr$	N/A	$2{\pi }^{2}R{r}^2$
Ellipse	$2\pi \sqrt{\frac{a^2+b^2}{2}}$	N/A	N/A
Parallelogram	$2(b+h)$	$bh$	N/A
Trapezium	$a+b+c+d$	$\frac{1}{2}(a+b)h$	N/A
Sector	$r^2\theta$	$r\times l$	$\frac{1}{2}{r}^2\theta$
Circular Segment	$r^2\theta -\frac{1}{2}{r}^2\sin (\theta )$	$r\times h$	N/A
Frustum of Cone	$\pi (R+r)(R-r)+\pi R^2+\pi r^2$	$\pi (R+r)l$	$\frac{1}{3}\pi h(R^2+Rr+r^2)$