EXERCISE . . Let f x ( ) = . Find the linear approximation at x = .
Use the linear approximation to approximate . . . Use the linear approximation to find approximate values of (i) ( (ii) (iii) .
Find a linear approximation for the following functions at the indicated points. (i) f x (ii) g x = − (iii) h x . The radius of a circular plate is measured as cm instead of the actual length . cm.
find the following in calculating change in the area of the circular plate: (i) Absolute error (ii) Relative error (iii) Percentage error . A sphere is made of ice having radius cm. Its radius decreases from cm to . cm.
Find approximations for the following: (i) change in the volume (ii) change in the surface area . The time T , taken for a complete oscillation of a simple pendulum with length l , is given by the equation T l g = p , where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of percent in the value of l . .
Show that the percentage error in the n th root of a number is approximately n times the percentage error in the number . . Differentials Here again, we use the derivative concept to introduce “ Differential ” . Let us take another look at ( ), df dx = lim lim ∆→ ∆→ + ∆ ∆ ′ ∆ ∆ ...( ) Here df dx is a notation, used by Leibniz, for the limit of the difference quotient, which is called the differential coefficient of y with respect to x .Will it be meaningful to treat df dx as a quotient of df and dx ?
In other words, is it possible to assign meaning to df and dx so that derivative is equal to and and