__Question 2:__

**True or False**. The derivative of a function at a given point gives the slope of the tangent line at that point.

__Answer :__

True. From the definition of the derivative.

__Question 3:__

**True or False**. If f ' is the derivative of f, then the derivative of the inverse of f is the inverse of f '.

__Answer :__

False. If g(x) is the inverse of f(x) then its derivative g '(x) is given by.

g '(x) = 1 / f ' (g(x)).

__Question 4:__

**True or False**. The derivative of ln a x, where a is a constant, is equal to 1 / x.

__Answer :__

True.

__Question 5:__

**True or False**. Rolle's theorem is a special case of the mean value theorem.

__Answer: __

True.

__Question 6:__

If f(x) = x^{ 3} -3x^{ 2} + x and g is the inverse of f, then g '(3) is equal to

(A) 10

(B) 1 / 10

(C) 1

(D) None of the above

__Answer :__

(B). Use g '(x) = 1 / f ' (g(x)) given as the answer to question 3 above to write g '(3) = 1 / f ' (g(3)).

First find g(3) which is the solution to the equation f(x) = 3 by definition of the inverse function.

x^{ 3} -3x^{ 2} + x = 3

The above equation has one real solution x = 3. So g(3) = 3, the solution of the above equation.

Then compute f '(x) = 3 x^{ 2} -6 x + 1.

f ' (g(3)) = 3 (3)^{ 2} -6 (3) + 1 = 10; and then substitute in the formula that gives g '(3) = 1 / 10.

__Question 7:__

**True or False**. The derivative of f(x) = a^{ x}, where a is a constant, is x a ^{ x-1}.

__Answer: __

False. Let y = a^{ x} so that ln y = x ln a

Differentiate both sides of ln y = x ln a with respect to x to obtain

(1 / y) dy / dx = ln a

Solve for dy / dx

dy / dx = y ln a = a^{ x} ln a

More references on calculus
questions with answers and tutorials and problems .