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2) Find a by ensuring that the slope of a straight line bears the correct relationship with f'(a), that is, with the slope of the curve y= f(x) at x = a. Question 3.13) (see boxed comment above) Is the following function differentiable at x =2? f(x) = { S+x² if x≤2, x-1 if x>2. Question 3.14) (see boxed comment above) Consider f(x)=2-x² and let g(x)=f(x). x= √2? A) Is g(x) differentiable at x= B) Provide two sketches of graphs: one for y= f(x) and one for y= g(x). Make sure that you show clearly when the functions are differentiable (smooth) and where they are non-differentiable (sharp).

2) Find a by ensuring that the slope of a straight line bears the correct relationship with f'(a), that is, with the slope of the curve y= f(x) at x = a. Question 3.13) (see boxed comment above) Is the following function differentiable at x =2? f(x) = { S+x² if x≤2, x-1 if x>2. Question 3.14) (see boxed comment above) Consider f(x)=2-x² and let g(x)=f(x). x= √2? A) Is g(x) differentiable at x= B) Provide two sketches of graphs: one for y= f(x) and one for y= g(x). Make sure that you show clearly when the functions are differentiable (smooth) and where they are non-differentiable (sharp).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you please answer question 3.14
For questions 3.11 to 3.14, you must use the definition of the derivative as a limit. Either use the formula "at a point" or "at any x" as appropriate. Do not use the differentiation rules seen afterwards. Question 3.11) In this question, use the formula for the slope at any x: f '(x) = lim f(x+h)-f(x) h h-o Of course, adapt this formula appropriately for different names of variables or functions. Do not change all variables to x, work with the given variable. Compute the derivative of: A) f(x)=x B) g(x)=1 D) f(x) = (2x-1)-¹ E) L(v)=√√c²- Question 3.12) (see boxed comment above) Consider the function f(x) = 2x². C) f(t)=2t-3t² where c is a constant. df A) Compute dx B) Find an equation for the line / which is tangent to the parabola y = f(x) and parallel to the line y-8x+100=0. C) Three lines passing through the point P(0,2) are normal (perpendicular) to the graph of y = f(x). One of them is the vertical line x=0 (the y-axis). Find the equation of the other two lines. Hints: 1) Requested lines will intersect y = f(x) at certain points Q. Denote the x-coordinate of such a point Q by a, so that Q(x, y) =Q(a, f(a)) = Q(a, 2a²). 2) Find a by ensuring that the slope of a straight line bears the correct relationship with f'(a), that is, with the slope of the curve y= f(x) at x =a. Question 3.13) (see boxed comment above) Is the following function differentiable at x = 2? [x² if x≤2, c)=√x-1 if x>2. 3 Question 3.14) (see boxed comment above) Consider f(x) = 2x² and let g(x)=f(x)|. x=√√2? A) Is g(x) differentiable at x = B) Provide two sketches of graphs: one for y = f(x) and one for y= g(x). Make sure that you show clearly when the functions are differentiable (smooth) and where they are non-differentiable (sharp). Homework Problems Page 6
Transcribed Image Text:For questions 3.11 to 3.14, you must use the definition of the derivative as a limit. Either use the formula "at a point" or "at any x" as appropriate. Do not use the differentiation rules seen afterwards. Question 3.11) In this question, use the formula for the slope at any x: f '(x) = lim f(x+h)-f(x) h h-o Of course, adapt this formula appropriately for different names of variables or functions. Do not change all variables to x, work with the given variable. Compute the derivative of: A) f(x)=x B) g(x)=1 D) f(x) = (2x-1)-¹ E) L(v)=√√c²- Question 3.12) (see boxed comment above) Consider the function f(x) = 2x². C) f(t)=2t-3t² where c is a constant. df A) Compute dx B) Find an equation for the line / which is tangent to the parabola y = f(x) and parallel to the line y-8x+100=0. C) Three lines passing through the point P(0,2) are normal (perpendicular) to the graph of y = f(x). One of them is the vertical line x=0 (the y-axis). Find the equation of the other two lines. Hints: 1) Requested lines will intersect y = f(x) at certain points Q. Denote the x-coordinate of such a point Q by a, so that Q(x, y) =Q(a, f(a)) = Q(a, 2a²). 2) Find a by ensuring that the slope of a straight line bears the correct relationship with f'(a), that is, with the slope of the curve y= f(x) at x =a. Question 3.13) (see boxed comment above) Is the following function differentiable at x = 2? [x² if x≤2, c)=√x-1 if x>2. 3 Question 3.14) (see boxed comment above) Consider f(x) = 2x² and let g(x)=f(x)|. x=√√2? A) Is g(x) differentiable at x = B) Provide two sketches of graphs: one for y = f(x) and one for y= g(x). Make sure that you show clearly when the functions are differentiable (smooth) and where they are non-differentiable (sharp). Homework Problems Page 6
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