Exponential function Definition and 161 Threads

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

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?

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...

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...

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

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 =...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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%...

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?

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} =...

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...

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...

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...

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...

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...

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?

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...

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

How does the exponential function work

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.

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)...

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...

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

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...

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}

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...

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...

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: [SIZE="4"]y'= (\frac{x^{lnx}}{x^2}) But it's not the...

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...

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

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.

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

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...

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)...

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

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...

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...

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?

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...

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)...

Homework Statement Use the function y=2e^-0.5x^2 to answer the following questions a) state the domain b) Determine the intercepts, if any c) Discuss the symmetry of the graph d) Find any asymptotes e)determine the intervals of increase and decrease f)what is the maxima and/or minima...

Homework Statement Determine the limit and then prove your claim. limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations I know that the formal definition that I need to use to prove the limit is: {limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...