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Question 3.24) A) Consider the function f(x)=√x. Find an equation for the line / which is tangent to the graph of y = f(x) and which is also perpendicular to the line k: 3x-y-10. B) Find an equation for the line which is tangent to the graph of y= g(x)=√x and which has y-intercept %. Hints: 1) The requested line will intersect y=g(x) at a certain point Q. Denote the x-coordinate of Q by a. 2) Find a > 0 ensuring that the slope of the straight line is the same as the derivative of y=g(x) at a.

Question 3.24) A) Consider the function f(x)=√x. Find an equation for the line / which is tangent to the graph of y = f(x) and which is also perpendicular to the line k: 3x-y-10. B) Find an equation for the line which is tangent to the graph of y= g(x)=√x and which has y-intercept %. Hints: 1) The requested line will intersect y=g(x) at a certain point Q. Denote the x-coordinate of Q by a. 2) Find a > 0 ensuring that the slope of the straight line is the same as the derivative of y=g(x) at a.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 35E
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Can you please answer question 3.24
3:49 < Back Homework 3.21 - 3.27 You are now expected to start using differentiation rules. Question 3.21) Differentiate the following. As always, work with the variables given, do not change them all to .x. A) f(x)=4x²-5x+√x+2 B) g(t)=5³+2cos(1)-1+√√F C) h(x)=(√x+x²) sin x D) g(0) tan 0 +1 Question 3.22) Show how one obtains the derivative of sec(0) using the known derivatives of sine and/or cosine. Simplify your answer to get it in the standard form involving secant and tangent. Question 3.23) A) Inspired by the proof of the Sum Rule done in class, prove the Difference Rule: If f(x) and g(x) are differentiable, then [f(x)-g(x)]'=f(x)-g'(x). B) Show how one obtains the derivative of cos(x) starting from the limit definition of the derivative. Adapt the procedure used for the derivative of sin(x) worked out in class. Question 3.24) A) Consider the function f(x)=√x. Find an equation for the line / which is tangent to the graph of y = f(x) and which is also perpendicular to the line k: 3x-y-10. B) Find an equation for the line which is tangent to the graph of y= g(x)=√x and which has y-intercept %. Question 3.25) Differentiate the following. (Chain rule required.) Simplify your answers when possible/nicer. A) r(x) = cos(10√x) B) g(t)=(-61² +6√1-3)* Hints: 1) The requested line will intersect y = g(x) at a certain point Q. Denote the x-coordinate of Q by a. 2) Find a>0 ensuring that the slope of the straight line is the same as the derivative of y= g(x) at a. E) r(x)= B) escl E) Question 3.26) Differentiate the following. (Chain rule required, sometimes more than once.) Simplify your answers when possible/nicer. A) f(x)=√4+sin' (%) D) sin(x)tan²(x) CSC (1) 2 sin x 1+xese.x cot(') 5(20-1)³ d [1 dx g(x) C) cos' e g'(x) [g(x)] D) tan(x)-tan'(x²) C) see' (√x) Question 3.27) Suppose some differentiable function g(x) is given, and consider its reciprocal f(x)=1/g(x). Use the definition of the derivative to prove the differentiation rule called the Reciprocal Rule: Homework Problems Page 7 Note: It may be handy to remember the Reciprocal Rule, although one may do calculus without it since it is equivalent to the use of other rules. More on this in Exercise Sets.
Transcribed Image Text:3:49 < Back Homework 3.21 - 3.27 You are now expected to start using differentiation rules. Question 3.21) Differentiate the following. As always, work with the variables given, do not change them all to .x. A) f(x)=4x²-5x+√x+2 B) g(t)=5³+2cos(1)-1+√√F C) h(x)=(√x+x²) sin x D) g(0) tan 0 +1 Question 3.22) Show how one obtains the derivative of sec(0) using the known derivatives of sine and/or cosine. Simplify your answer to get it in the standard form involving secant and tangent. Question 3.23) A) Inspired by the proof of the Sum Rule done in class, prove the Difference Rule: If f(x) and g(x) are differentiable, then [f(x)-g(x)]'=f(x)-g'(x). B) Show how one obtains the derivative of cos(x) starting from the limit definition of the derivative. Adapt the procedure used for the derivative of sin(x) worked out in class. Question 3.24) A) Consider the function f(x)=√x. Find an equation for the line / which is tangent to the graph of y = f(x) and which is also perpendicular to the line k: 3x-y-10. B) Find an equation for the line which is tangent to the graph of y= g(x)=√x and which has y-intercept %. Question 3.25) Differentiate the following. (Chain rule required.) Simplify your answers when possible/nicer. A) r(x) = cos(10√x) B) g(t)=(-61² +6√1-3)* Hints: 1) The requested line will intersect y = g(x) at a certain point Q. Denote the x-coordinate of Q by a. 2) Find a>0 ensuring that the slope of the straight line is the same as the derivative of y= g(x) at a. E) r(x)= B) escl E) Question 3.26) Differentiate the following. (Chain rule required, sometimes more than once.) Simplify your answers when possible/nicer. A) f(x)=√4+sin' (%) D) sin(x)tan²(x) CSC (1) 2 sin x 1+xese.x cot(') 5(20-1)³ d [1 dx g(x) C) cos' e g'(x) [g(x)] D) tan(x)-tan'(x²) C) see' (√x) Question 3.27) Suppose some differentiable function g(x) is given, and consider its reciprocal f(x)=1/g(x). Use the definition of the derivative to prove the differentiation rule called the Reciprocal Rule: Homework Problems Page 7 Note: It may be handy to remember the Reciprocal Rule, although one may do calculus without it since it is equivalent to the use of other rules. More on this in Exercise Sets.
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