2.3 Differentiabilityap Calculus

1. 2.3 Differentiabilityap Calculus Calculator
2. 2.3 Differentiabilityap Calculus Notes
3. 2.3 Differentiabilityap Calculus Test
4. 2.3 Differentiabilityap Calculus Definition

Program logitech curve zoom button for mac. 2 3 lim x 37 x →∞xx + =∞ −+ This function has end behavior like x3 5 2 x x ⎛⎞ ⎜ ⎝⎠ ⎟. The function does not reach a limit, but to say the limit equals infinity gives a very good picture of the behavior. If the x with the largest exponent is the same, numerator and denominator, the limit.

1. CHAPTER 2 Differentiation Section 2.1 The Derivative and the Tangent Line Problem 95 1. (a) At slope At slope (b) At slope At slope x2, y 2, 2.
2. Differentiability and Differentiation - Mathematics - AP EAMCET Engineering - Online Practice Test MCQ - 2020. Subscribe and get the complete access of Mathematics - AP EAMCET Engineering MCQ Online Test - MCQ's Online Test Series Pack and score high marks.

Example 1:

Show that the function has a solution between 2 and 3.

Solution:

Plugging in 2 and 3 into f(x), we see that f(2) = ln(2) 1 -0.307 and f(3) = ln(3) 1 0.099. Since the first number is negative and the second number is positive and f(x) is a continuous function on the interval [2, 3], by the Intermediate Value Theorem, f(x) must have a solution between 2 and 3.

Show that the function has only one real solution.

Solution:

First we use the Intermediate Value Theorem to show that there is at least one solution. We can use the theorem since f(x) is a continuous function everywhere. Notice that f(1) = -1 and f(2) = 45. That means that somewhere between 1 and 2, f(x) = 0.

Well, we have shown that there is at least one solution to the equation. Now we have to show that it is the only solution. To do that, we shall show that it is not possible for f(x) to have a second solution.

We do this by taking the derivative. . Notice that . That means that f(x) is a strictly increasing function. A strictly increasing function will only have one x-intercept (solution), and thus f(x) has only one real solution.

Problems For You To Solve

Multiple Choices:

1. In what situation f'(x) might NOT fail to exist？

A. cuspe

B. conner

C. jump

D. continuous

E. IDK

2. Let f be a continuous function on the closed interval [-3.6]. If f (-3) =- 1 and f (6) = 3, then the Intermediate Value Theorem guarantees that

A. f(0)=0

Transfer apps from ipad to mac prosalenew. B. for at least one c between -3 and 6

C. for all x between -3 and 6

D. f(c)=1 for at least one c between -3 and 6

E. f(c)=0 for at least one c between -1 and 3

3. The function f is continuous on the closed interval [0, 2] and has values that are given in the table above. The equation must have at least two solutions in the interval [0, 2] if k=

A. 0

B. 1

C. 2

2.3 Differentiabilityap Calculus Calculator

D. 3

E. 4

4. Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the following must be true?

I. f has at least 2 zeros.

II. The graph of f has at least one horizontal tangent.

III. For some c, 2 < c < 5 , f(c)=3.

A. None

B. I only

C. I and II only

D. I and III only

E. I, II and III

5. The function f is continuous for and differentiable for -2 < x < 1. If f(-2) =-5 and f(l) = 4, which of the following statements could be false?

A. There exists c, where -2 < c < 1, such that f (c) = 0.

B. There exists c, where -2 < c < 1, such that f '(c) = 0.

C. There exists c, where -2 < c < 1, such that f(c) =3.

D. There exists c, where -2 < c < 1, such that f '(c) = 3.

6. Let f(x) = x|x| and g(x) =sin x. Assertion : gof is differentiable at x = 0 and its derivative is continuous at that point Reason : gof is twice differentiable at x = 0.

2.3 Differentiabilityap Calculus Notes

(1) Find the derivatives of the following functions using first principle.

(i) f(x) = 6Solution

(ii) f(x) = -4x + 7Solution

(iii) f(x) = -x² + 2 Solution

(2) Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

(i) f(x) = |x - 1|Solution

(ii) f(x) = √(1 - x²) Solution

(3) Determine whether the following function is differentiable at the indicated values.

2.3 Differentiabilityap Calculus Test

(i) f(x) = x | x | at x = 0 Solution

(ii) f(x) = |x² - 1| at x = 1 Solution

(iii) f(x) = |x| + |x - 1| at x = 0, 1 Solution

(iv) f(x) = sin |x| at x = 0 Solution

(4) Show that the following functions are not differentiable at the indicated value of x.

(i)

Solution

(5) The graph of f is shown below. State with reasons that x values (the numbers), at which f is not differentiable.

Solution

(6) If f(x) = |x + 100| + x², test whether f'(-100) exists.

(7) Examine the differentiability of functions in R by drawing the diagrams.

(i) | sin x | Solution

(ii) |cos x| Solution

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