MATH 110 Lecture 39: Derivative of the Exponential Function

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cerisegiraffe538

2 Nov 2015

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The derivative of the exponential function with base e is equal to the function itself: ed dx x = ex using the limit definition of the derivative. ))/h (e (e lim h 0 h 1. Ex. g(x) g (x) d dx x+h 3x. = lim h 0 lim h 0 h 1 (3 (3 lim h 0. = x2 x e x x + x2 d e dx. = ( * ex x * ex. Find critical points of (x) (x )e f (x) e (2.

Related Questions

Many airlines restrict the size of carry on item in terms of theircombined length, width and height (L+W+H).
I am due to fly on an airline that states "The sum of the length,the width and height of your item must be at most 70 inches".
Suppose I want to take a box (cuboid) on my trip that will allow meto carry on as much as possible, and that I also want this box tohave a square base.
The questions we will answer using optimization are: Whatdimensions should the box have, and what would be the resultingmaximum volume

a.If the base of the box is a square whose sides have length x, andthe height of the box is h (both measured in inches), enterexpressions for the volume of the box, V, and the sum of thelength, width and height of the box in terms of x and h.
V :

Sum L+W+H :

b.Since the sum of the length, width and height must be less thanor equal to 70 inches, and it is clearly best to choose this sum toequal 70 inches, write down the constraint equation.
Rearrange the constraint equation to give the height of the box, h,in terms of the length of the base, x

h =

c.What is the objective function for this problem?
Using the constraint equation, rewrite the objective function interms of x alone

V =

d.Differentiate V with respect to x, to find the derivative dV /dx.
=

e.Find the values of x for which we have a potential relativeextreme point of V.

One of these is at x = 0, and the other occurs when x is equal towhat value?

(Give your answer, and those that follow, correct to two decimalplaces.)

x =

For this question, you need not show that this is actually arelative maximum point. (But you should know how you would dothis.)

f.Now that you know the length of the base of the box, what is itsheight?
height : inches

g.What is the maximum volume of the box?
volume : cubic inches

1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

MATH 110 Lecture 39: Derivative of the Exponential Function

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