Mean Value Theorem Practice Problems

Introduction to mean value theorem practice problems:

Let f: [a,b] -> R be a continuous function on [a,b] and differentiable on (a, b). Then the Mean value theorems states that there exists some c in (a,b) such that

f '(c) = [f(b) - f(a)] / (b-a)

This is the one of the most fundamental theorem and result in Calculus. The geometrical interpretation of the mean value theorem is that the slope of tangent drawn at (c, f(c)) is same as the slope of the secant between (a, f(a)) and (b, f(b)). In other words , there is a point c in (a,b) such that the tangent at (c,f(c)) is parallel to the secant between (a,f(a)) and (b,f(b)).

Here are some of the mean value theorem practice problems.

Mean Value Theorem Practice Problem and its Solution.

Problem: Verify Mean value theorem for the function f(x) = x2 in the interval [2,4].

Solution: The function f(x) = x2 is continuous in [2,4] and differentiable in (2,4) as its derivative f '(x) = 2x is defined in (2,4).

Now f(2) = 4 and f(4) = 16. Hence

[f(b) - f(a)] / (b-a) = (16-4)/(4-2) = 6

Mean value theorem states that there is a point c belongs to (2,4) such that f '(c) = 6. But f '(x) = 2x which implies c= 3 and this belongs to (2,4), therefore f '(c) =6. Between, if you have problem on these topics Decimal Notation, please browse expert math related websites for more help on Discount Formula.

Practice Problems-mean Value Theorem :

Problem: Verify Mean Value Theorem, if f(x) = x2-4x -3 in the interval [a,b], where a=1 and b=4.

Problem: Examine the applicability of Mean value theorem for the function f(x) = x2 - 1 for x belonging to [1,2].

Problem: Verify Mean Value Theorem, if f(x) = x3-5x2 -3x in the interval [a,b], where a=1 and b=3. Find all c belonging to (1,3) for which f '(c) =0.

Problem: Verify the Mean Value Theorem for the function f(x) = x2 +2x -8 for x belonging to [-4,2].

Problem: Verify the Mean Value Theorem for the function f(x) = (x-4) (x-6) (x-8) in [4,10].

Problem: Prove the Mean Value Theorem for the function f(x) = 1/x for the interval [-1, 1].

Problem: Examine if the Mean value theorem is applicable or not to the function f(x) = [x] for x belonging to [5,9] where [x] represent the greatest integer function.