Determine the parameters a, b, c in the equation y = ax² + bx + c so that it becomes tangent to the line y = x at the point x = 1 and passes through the point (−1, 0).

Prove that the curve y₁ = f(x) (where f(x) > 0) and y₂ = f(x)sin ax where f(x) is a differentiable function, are tangent to each other at the common points.

How will the expression
 
be transformed (where y is twice differentiable function of x) if we introduce a new variable t, putting x = cost.

In determining an electric current by means of a tangent galvanometer use is made of the formula l = k tan Φ where l = current, k = factor of proportionality (depending on the instrument), Φ = angle of pointer’s deflection. Determine the relative error of the result which depends on the inaccuracy in reading the angle Φ. At what position of the pointer can we obtain the most reliable results?

Derive the following formula using constancy [Definition of constancy – If at all points of an interval (x) = 0 then the function f (x) is a constant within the interval]
 

Derive the following formula using constancy [Definition of constancy – If at all points of an interval (x) = 0 then the function f (x) is a constant within the interval]
 

Derive the following formula using constancy [Definition of constancy – If at all points of an interval (x) = 0 then the function f (x) is a constant within the interval]
 

Prove that the square root of two successive natural numbers greater than differ by less than .

Applying L’Hospital’s rule, find the limit of the function
 

Applying L’Hospital’s rule, find the limit of the function
 

Find the limit
 

Find the limit
 

Find the limit
 

Find the limit
 

Find the limit
 

Find the limit
 

Find the limit
 

Find the limit
 

Compute
 

Ascertain the existence of the limits
 
Can L’Hospital’s rule be applied in computing them? Does the formal application lead to the correct answer?

Using L’Hospital’s rule find the limit of the following function
 

Using L’Hospital’s rule find the limit of the following function
 

Using L’Hospital’s rule find the limit of the following function