Advertisements
Advertisements
Question
Examine the function
f(x) `{:(= x^2 cos (1/x)",", "for" x ≠ 0),(= 0",", "for" x = 0):}`
for continuity and differentiability at x = 0
Solution
f'(0) = `lim_("h" -> 0) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0) ("f"("h") - 0)/"h"`
= `lim_("h" -> 0) ("h"^2 cos (1/"h"))/"h"`
= `lim_("h" -> 0) "h" cos(1/"h") ...[(because "h" -> 0),(therefore "h" ≠ 0)]`
= `[lim_("h" -> 0) "h"] xx [lim_("h" -> 0) cos(1/"h")]`
= [0] × [FInite number between – 1 and 1]
= 0
∴ f is differentiable at x = 0 and hence continuous at x = 0.
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following function w.r.t. x.:
x–9
Find the derivative of the following function w. r. t. x.:
`7xsqrt x`
Differentiate the following w. r. t. x. : `x sqrtx + logx − e^x`
Differentiate the following w. r. t. x. : `2/7 x^(7/2) + 5/2 x^(2/5)`
Differentiate the following w. r. t. x. : x3 log x
Differentiate the following w. r. t. x. : ex log x
Find the derivative of the following w. r. t. x by using method of first principle:
x2 + 3x – 1
Find the derivative of the following w. r. t. x by using method of first principle:
e2x+1
Find the derivative of the following w. r. t. x by using method of first principle:
3x
Find the derivative of the following w. r. t. x by using method of first principle:
log (2x + 5)
Find the derivative of the following w. r. t. x by using method of first principle:
sec (5x − 2)
Find the derivative of the following w. r. t. x by using method of first principle:
`x sqrt(x)`
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`sqrt(2x + 5)` at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
tan x at x = `pi/4`
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
log(2x + 1) at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`"e"^(3x - 4)` at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
cos x at x = `(5pi)/4`
Show that the function f is not differentiable at x = −3, where f(x) `{:(= x^2 + 2, "for" x < - 3),(= 2 - 3x, "for" x ≥ - 3):}`
Discuss the continuity and differentiability of f(x) = x |x| at x = 0
Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`
Discuss the continuity and differentiability of f(x) at x = 2
f(x) = [x] if x ∈ [0, 4). [where [*] is a greatest integer (floor) function]
Select the correct answer from the given alternative:
If y = `("a"x + "b")/("c"x + "d")`, then `("d"y)/("d"x)` =
Select the correct answer from the given alternative:
If, f(x) = `x^50/50 + x^49/49 + x^48/48 + .... +x^2/2 + x + 1`, thef f'(1) =
Determine the values of p and q that make the function f(x) differentiable on R where
f(x) `{:( = "p"x^3",", "for" x < 2),(= x^2 + "q"",", "for" x ≥ 2):}`
Determine all real values of p and q that ensure the function
f(x) `{:( = "p"x + "q"",", "for" x ≤ 1),(= tan ((pix)/4)",", "for" 1 < x < 2):}` is differentiable at x = 1
Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R
Test whether the function f(x) `{:(= 2x - 3",", "for" x ≥ 2),(= x - 1",", "for" x < 2):}` is differentiable at x = 2
If y = `"e"^x/sqrt(x)` find `("d"y)/("d"x)` when x = 1
