Calculus Answers

Take 10 figure logarithm to base 10 from x = 300 to x = 310 by unit increments. Calculate

the first derivative of log10 x when x = 310.

y=3z+4x^2

When a pendulum of length 10 cm has swung so that 𝜃 is the radian measure of the angle formed by the pendulum and a vertical line, then if ℎ(𝜃) cm is the vertical height of the end of the pendulum above its lowest position, ℎ(𝜃) = 20 𝑠𝑖𝑛^2 𝜃/2 . Find the instantaneous rate of change of ℎ(𝜃) with respect to 𝜃 when (a) 𝜃 = 𝜋/3 and (b) 𝜃 = 𝜋/2

Use Rolle's theorems to show that there is a solution of the equation cotx=x in ]0,π/2[

Suppose that g(3)=1, g'(3)=2 and f'(-1)= 0.5 find f(g(x))'3

using suitable function f(x), obtain the following inequality for a < b < 1 using mean value theorem

A particular company has determined that when its weekly advertising expense is x dollars, then if S dollars is its total weekly income from sales, S =4000 lnx.

(a) Determine the rate of change of sales income with respect to advertising expense when 800 is the weekly advertising budget.

(b) If the weekly advertising budget is increased to $950 dollars, what is the approximate increase in the total weekly income from sales?

Differentiate the given functions.

1. g(x)=(1)/(2)sec^(-1)(4)/(x)+2csc^(-1)4x
2. (x)=6sin^(-1)\sqrt(1-x^(2))
3. h(x)=xsin^(-1)2x
4. y=sec^(-1)\sqrt(x^(2))+4
5. y=cos^(-1)(sinx)

f(x)=6sin^(-1)\sqrt(1-x^(2))

Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) = 5 for all (x, y) ∈ R2 with x^2+y^2<1