Exponential function Definition and 166 Threads

  1. C

    MHB Implicit differentiation with exponential function

    find dy/dx: exy+x2+y2= 5 at point (2,0) I'm confused with finding the derivative with respect to x of exy. this is what I did so far for just this part: exy*d(xy)/dx exy*(y+x*dy/dx) do I need to put the parentheses on here? I thought so because that is the part where I used the product rule...
  2. Y

    Is j^{-p}=e^{-j\frac{p\pi}{2}} a valid exponential function?

    Homework Statement I want to verify j^{-p}=e^{-j\frac{p\pi}{2}} Homework Equations e^{j\frac{\pi}{2}}=\cos(\frac{\pi}{2})+j\sin(\frac{\pi}{2})=j The Attempt at a Solution j^{-p}=(e^{j\frac{\pi}{2}})^{-p}=e^{-j\frac{p\pi}{2}} Am I correct? Thanks
  3. C

    MHB Find derivative with exponential function?

    f(x)=x2ex the answer is f'(x)=(x2 + 2x)ex but I don't understand how to get there. Also I need to find g'(x) if g(x)=sqrtx(ex) would the answer for the second one be .5x-1/2ex?
  4. paulmdrdo1

    MHB Integration of exponential function

    just want to confirm if i did set up my integral correctly and got a correct answer. $\displaystyle\int_0^a (e^{\frac{x}{a}}-e^{-\frac{x}{a}})$ using substitution for the first term in my integrand $\displaystyle u=\frac{x}{a}$ $\displaystyle du=\frac{1}{a}dx$; $\displaystyle dx=adu $ for...
  5. Drakkith

    Evaluating an exponential function that models a real-world situation

    Homework Statement Suppose that the velocity v(t) (in m/s) of a sky diver falling near the Earth's surface is given by the following exponential function, where time is measured in seconds. v(t) = 55 (1-e-0.18(t)) Find the initial velocity of the sky diver and the velocity after 6...
  6. MarkFL

    MHB Chris' question at Yahoo Answers regarding an exponential function

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  7. C

    Solve for x in exponential function

    Homework Statement So, i am given 3^x(2x) = 3^x + 2x + 1 And i want to solve for x. Homework Equations I only know that the solution is x=1 but i don't know how to get there. The Attempt at a Solution 3^x(2x) = 3^x + 2x + 1 3^x(2x) - 3^x = 2x + 1 3^x(2x - 1) = 2x + 1 3^x =...
  8. C

    Second derivative of an exponential function

    Homework Statement I have the derived function: f'(x) = [1/(1+kx)^2]e^[x/(1+kx)] k is a positive constant Homework Equations I need to find the second derivative, which I thought was just the derivative of the exponent multiplied by the coefficient (as you find the...
  9. L

    On the Expansion of Exponential Function by Integration

    I ask members here kindly for their assistance. I'm having some confusion over the process of integrating inequalities, in particular for obtaining the series expansion for the exponential function by integration. The text by Backhouse and Holdsworth (Pure Mathematics 2), shows the expansion of...
  10. B

    MHB Limit involving exponential function

    Hello everyone, how are you? I'm having trouble to evalue the following limit: \lim_{x->\infty} (\frac{x}{1+x^2})^x I "transformed" it into e^{ln{(\frac{x}{1+x^2})^x}} and tried to solve this limit: \lim_{x->\infty} x ln{(\frac{x}{1+x^2})} But I have no idea how to solve it correctly. Can...
  11. P

    Exponential function and chain rule - find derivative

    Homework Statement If f(x) = e^{3x^2+x} , find f'(2)Homework Equations f'(x) = a^{g(x)}ln a g'(x)The Attempt at a Solution f'(x) = (e^{3x^2+x})(ln e)(6x+1) f'(2) = (e^{3(2)^2+2})(ln e)(6(2)+1) = 2115812.288 I was checking online and I'm seeing a different answer, but this is EXACTLY how...
  12. M

    Finding the magnitude of a complex exponential function

    Homework Statement I want to know the steps involved in finding the magnitude of a complex exponential function. An example of the following is shown in this picture: Homework Equations |a+jb|=sqrt(a^2+b^2) |x/y|=|x|/|y| The Attempt at a Solution For the denominator, I replaced z with e^jw...
  13. L

    Why Does exp(-z^2) Approach Zero in Certain Sectors?

    Homework Statement Reading Hinch's book, there is a statement as follows: ... z need to be kept in the sector where exp(-z^2) ->0 as z -> infinity. Thus it's applicable to the sector |arg z|<pi/4...Homework Equations Why is this true and what is the limiting behavior of exp(x) for x in...
  14. B

    MHB Quick modulus question - complex exponential function

    What is |exp (z^n)| less than if |z| < 1? I'm thinking it's e but I'm having a brain freeze at the moment! Thanks for any help guys.
  15. J

    Understanding Polar Coordinates and the exponential function

    I'm reviewing math material for the EIT exam, I'm going over math concepts that should be pretty basic but I feel like there are gaps in my understanding. I understand how we can use rectangular coordinates and complex numbers to find a point on the complex plane. It would follow logically...
  16. P

    Scatter plot, equation of exponential function

    Homework Statement The copy of the question is in the image attached. I don't have a graphing calculator, and so am not sure how to determine the equation of the exponential function. Homework Equations y = a^x The Attempt at a Solution I made a scatter plot off of an online...
  17. A

    Simple formula for an exponential function. Please help i'm stuck

    Q A cold yam is plaveedd in a hot oven. Newton's Law of Heating tells us that the difference between the oven's temperature and the yam's temperature decays exponentially with time. The yam's temperature is initially 0 deg F, the oven's temp is 300 deg F, and the temp difference decreases by 3%...
  18. L

    MHB Help with exponential function word problem

    The number of bacteria in a culture is given by B(t) = 40e0.6t, where t is the time in days. How many bacteria are there after 2 days? I substituted t = 2, and got this: 2B = 40e1.2 which I simplified to become this B = 66.4 days Apparently this answer is wrong though, can someone explain why?
  19. Q

    Why do we use the natural log in the derivative of an exponential function?

    I recently struck a question that I have not been able to find an answer to. I feel like I'm missing something obvious, so I've come here for help. The derivative of a^{x} is a^{x}lna. The explanation that Stewart 5e gives is: \frac{d}{dx}a^{x} = \frac{d}{dx}e^{(lna)x} =...
  20. M

    Exponential Function Homework: Showing 0 ≤ e^x−1−x

    Homework Statement k is a real number and f_{k}(t)=e^{t}-1-t-k\frac{x^{2}}{2} 1- Show that : (\forall x\epsilon\mathbb{R}):0\leq e^{x}-1-x 2- Show that : (\forall x>0)(\exists k\epsilon\mathbb{R}^{+})(\exists d\epsilon[0,x]):f(x)=f''(d)=0 3-Conclude that (\forall...
  21. D

    Derivative of exponential function

    Homework Statement Find extrema and points of inflection (then graph it). f(x) = x2e-x Homework Equations The Attempt at a Solution So, for f'(x) = xe-x(2-x) Critical point(s): f'(x) = 0 2-x = 0 x= 2 I have a question before continuing. Will my critical point include 0 besides 2? I...
  22. C

    How Do You Invert the Function Q(t) in a Camera Flash Capacitor Model?

    Homework Statement When a camera flashes, the batteries begin recharging the flash capacitor which stores the charge Q according to the function Q(t) = Q* (1-e-t/a) where t is the elapsed time in seconds since the camera flash and Q* and a are non-zero (a) What does Q* represent? (b) Find the...
  23. P

    Very simple exponential function quesiton

    The derivative of e^(2x): let y = e^(2x), let u = 2x, so y = e^u chain rule: du/dx * dy/du = 2*e^u = 2e^u = 2e^(2x) this is the solution copied from my book, my question is why do they let u = 2x? is e^u the same as e^x? If so then wouldn't all derivatives of the exponential functions be in...
  24. A

    Spivak - Radioactivity and the Exponential Function

    Homework Statement A radioactive substance diminishes at a rate proportional to the amount present (since all atoms have equal probability of disintegrating, ...). If A(t) is the amount at time t, this means that A'(t)= p * A(t) for some p representing the probability that an atom will...
  25. B

    Derivative of Exponential Function

    I am reading the about the derivative of an exponential function using the limit definition, but one step I don't quite understand: lim_{h\rightarrow0}\frac{a^h -1}{h} = f'(0) Wouldn't that limit equal 0/0?
  26. Y

    Derivative of the exponential function with base e

    For an exponential function of the form y=a^x First derivative , d/dx [a^x ] = a^x∙ lim┬(δx→0)⁡{(a^δx-a^0)/δx} = a^x∙ m_((0,1)) now if m_((0,1)) which is the gradient of the y-axis intersection point of the exponential function equal to 1 exactly...
  27. C

    Exponential function differentiation

    if first derivative is the slop of the given functions, then what is the physical meaning of exponential function remaining the same function after differentiation?? does it mean its vertical tangency make it indifferentiable? please clarify me the concept... regards
  28. M

    How does the exponential function work?

    How does the exponential function work
  29. P

    What is the Correct General Expression for an Exponential Function?

    Hi, I am working on a modeling exercise and I would like to know which is the correct general expression for a Exponential Function: y = abc(x+d)+e or y = abcx+d+e Thanks in advance, Peter G.
  30. V

    Derivation of imaginary exponential function

    I would like to find derivations of exp(-ik0r) respect to k in order to calculate exp(-ik1r) by using Taylor expansion: exp(-ik1r) = (exp(-ik0r))(0) +(k1 -k0)(exp(-ik0r))(1)/1! + (k1 -k0)2(exp(-ik0r))(2)/2! + ... This expansion converges when the value of r is relative low (0.3 - 1.2)...
  31. N

    Derivative of an exponential function using logarithms (lon-capa)

    Homework Statement What is the derivative of y=x^(13/x^2) with respect to x? The Attempt at a Solution I went through multiple techniques to solve this, but all of them have failed so far ._. In my latest attempt, I took the natural log of both sides: lny= lnx^(13/x^2) I...
  32. E

    Using L'hopital's rule in exponential function

    Is that any way to find a finite value which is not equal to zero using L'hopital's rule in limit(z=-ia) exp[-A/(z+ia)]/(z+ia)^2 i kept getting 0/0 after differentiation Thank you
  33. R

    Exponential function sum problem

    Homework Statement I need to prove that (1+x_{1})·...·(1+x_{n})≥(1-Ʃ^{n}_{i=1}x_{i}^2)e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 I've already proven that (1+x_{1})·...·(1+x_{n})≤e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 and (1-x_{1})·...·(1-x_{i})≥1-Ʃ^{n}_{i=1}x_{i} with all 0≤x_{i}≤1 , but...
  34. R

    Exponential function proof problem

    Homework Statement Big problem with exponential function proof assignment, need some help. let x≥0 and for every k\in N there is n_{k}\in N and x_{k1}≥...≥x_{k_{nk}} and x_{k1}+...+x_{k_{nk}}=x. Proof: if lim_{k→}∞ x_{k1}=0 then lim_{k→}∞ (1+x_{k1})·...·(1+x_{k_{nk}})=exp(x)=e^{x}
  35. C

    How Does the Natural Exponential Function Influence Trigonometric Outcomes?

    1. Let f(x)=sin(e^x) a. Find 2 values of x satisfying f(x)=0 b. What is the range of f(x) c. Find the value(s) of x that maximize f on [3.8,4] (use calculus) 2. y=e^x if and only if x=ln y 3.a. x=-infinity because the limit of e^x as x approaches -infinity is 0. and also...
  36. Z

    Exponential function with 3 parameters?

    Exponential function with 3 parameters?? Homework Statement A researcher suggests that the population, P, at time t can be modeled by P(t) = K / (1 + Le^(-Mt)), where K, L and M are parameters. Use technology to estimate and interpret K, L,and M. Construct the researcher's model using...
  37. I

    How Do You Differentiate y = x^(ln x)?

    Homework Statement Find dy/dx. y=x^{lnx}Homework Equations The Attempt at a Solution ln y= ln x^{ln x} ln y= (ln x)(ln x) Taking the derivative now: \frac{1}{y}y'= (\frac{1}{x})(\frac{1}{x}) \frac{1}{y}y'= (\frac{1}{x})^2 Multiply by y: y'= (\frac{x^{lnx}}{x^2}) But it's not the right...
  38. E

    Tricky Fourier Transform problem for an exponential function

    Homework Statement find the Fourier transform, using the definition of the Fourier transform \widehat{f}(\nu)=∫^{∞}_{-∞}f(t)e^{-2 \pi i \nu t}dt, of the function f(t)=2 \pit^{2}e^{- \pi t^{2}}Homework Equations I have the answer: (1-2{\pi \nu^{2}})e^{- \pi \nu^{2}} The Attempt at a Solution...
  39. W

    Exponential function integral with Sin as its index

    Integral descriptive: \int _{0}^{\pi }\! \left( a+{\it k1}\,\sin \left( \theta+{\it beta1} \right) \right) {e}^{{\it k2}\,\sin \left( \theta+{\it beta2} \right) }{d\theta} I think if beta1 and beta2 come to zero,i may be a typical integral.However,in this integral,beta1 and beta2 are...
  40. A

    How to calculate the convergence point for exponential function?

    Homework Statement I need to calculate the point of divergence for this exponential function : F(x)= 5.282 * exp ( -0.01726 * x ) may you help me in finding the method to solve such problems ? Homework Equations The Attempt at a Solution
  41. S

    Solve Problem Algebraically: f{g(x)} and g{f(x)}

    f(x)= lnx ; g(x) = ex f{g(x)} = ln ex = x; not an issue g{f(x)}= eln x = ? (answer for this) f(x) and g(x) are inverse of each other. how to solve the problem algebraically.
  42. S

    Exponential function and Geometric progression

    Can anyone help me answer this question? " Every exponential function is a geometric progression but not every geometric progression is an exponential function. Explain."
  43. O

    About the exponential function of series

    hey, I have a question about the taylor series.I open cosθ function, I forward while making this solution but I can not write as the definition of exponential function. How Can I write this expression as an exponential function. Maybe I can not see something in it.. Do you have any idea...
  44. B

    Why is the sum of exponential function 2^x for n iterations one less than 2^n?

    I posted a related thread, in the same forum regarding matlab, but I want to discuss this here separately. Few minutes ago I became very confused baffled and surprised at the same time. consider function 2^{x}. Now consider the sum of this function for n iterations.(from 0 to n)...
  45. S

    Exponential Function: Understanding and Solving Problems

    Help! exponential function Please look at the attachment for the question! Thank you for your help
  46. M

    Inverse of exponential function

    Homework Statement Give the inverse of this function N=f(L)=1816-8L The answer has to be filled in in Maple Homework Equations The Attempt at a Solution N=1816-8L 16-8L=ln(N)/ln(18) -8L=(ln(N)/ln(18))-16 L=-ln(N)/(8*ln(18))+2 Is this correct? When i fill...
  47. A

    Simple exponential function exercise

    Homework Statement is e^x^2 = 4 equivalent to e^x = 2 Homework Equations As above The Attempt at a Solution This is just an exercise, but I'm quite stuck as how to show this is true (or false for that matter). I thought to take the log of both sides and use the log identity to get...
  48. JeremyEbert

    When Does the Exponential Function Reach Its Maximum Value?

    why is it that the largest value of n^( (((n-k)*(k-1)/(2k)) + (k-1))/(n-1) ) always seems to be when k=36?
  49. mccoy1

    Approximation involving an exponential function

    Homework Statement I was following a derivation of some laws and I didn't get how they approximate some portion of the expression. That portion/part is exp[gbH/(2kT)]. The book says gbH/2 <<1 and therefore exp[gbH/(2kT)] = 1+gbH/(2kT). Homework Equations The Attempt at a Solution I agree with...
  50. A

    Integration of exponential function times polynomial of fractional degree

    Homework Statement I'm working out a differential equation problem that I am supposed to solve with the formula \mathcal{L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}. The problem is \mathcal{L}\{t^{\frac{1}{2}}\} (finding the Laplace transform of the given function)...
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