Calculus Ch. 4. The Derivative
Question 1 (October 2011 Exam, Q 12). Complete the following definition of A function f (x) has a right derivative A at a point x if such that implies Answer: b)
Question 2 (October 2011 Exam, Q 20). Let Find an approximate value for using differentials. Solution. Use the following approximation where x 0 = 1, and We have Hence Answer: a).
Tangent lines. 5 1 x
Question 3 (November 2008 Exam, Q 14). At which of the following points is the tangent line drawn to the graph of f (x) = x 3 + 4 x 2 - 10 parallel to the line y = 20 - 4 x? Solution. The equation of the tangent line at a point x = x 0: The two lines are parallel if We have The equation 3 x 2 + 8 x = -4 has solutions Answer: b.
Normal lines. 5 1 x
Question 4 (November 2008 Exam, Q 17). What is the equation of the line normal to the graph of f (x) = exp(x - 2) at the point x = 2? Solution. The equation of the normal line at a point x = x 0: We have Hence for x 0 = 2, and Answer: c.
Derivative of inverse functions f -1(x). Differentiate both sides of this equation using the chain rule. Hence
Question 5 (November 2009 Exam, Q 23). What it the derivative of the inverse function at x = 3, if Solution. We have and It is necessary to find Since Hence we obtain
Derivative of implicit functions. We are given an equation If we solve this equation for y we obtain a function Can we find the derivative without solving the equation
Question 6 (November 2009 Exam, Q 22). Find if y(1) = 2 and y(x) is an implicit function given by Solution. We have and =1 Hence =1 =1 =1
The Lagrange (Mean Value) Theorem. Let f (x) be a function which satisfies the following hypothesis: 1. f (x) is continuous on the closed interval [a, b]. 2. f (x) is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that
Question 6. Suppose that f (0) = 1 and for all values of x. How small f (2) can possibly be? Solution. According to the Lagrange theorem Hence Answer: The smallest possible value for f (2) is 3.
When the goings get tough the tough get going!