Calculus Question

Question 1 Limits

Evaluate the limit:

lim_{x to 0} frac{e^{3x} – 1 – 3x}{x^2}

Answer:

Use the Taylor expansion:

e^{3x} = 1 + 3x + frac{9x^2}{2} + cdots

frac{e^{3x} – 1 – 3x}{x^2} = frac{frac{9x^2}{2} + cdots}{x^2} to frac{9}{2}

Final Answer: boxed{frac{9}{2}}

Question 2 Differentiation

Find the derivative:

f(x) = x^x

Answer:

Use logarithmic differentiation:

ln f = x ln x

Differentiate:

frac{f’}{f} = ln x + 1

f’ = x^x(ln x + 1)

Final Answer: boxed{x^x(ln x + 1)}

Question 3 Optimization

Find the maximum value of

f(x) = x e^{-x}, quad x ge 0

Answer:

Differentiate:

f'(x) = e^{-x} – xe^{-x} = e^{-x}(1-x)

Set to zero x=1

f(1)=frac{1}{e}

Maximum value: boxed{frac{1}{e}} at x=1

Question 4 Integration

Evaluate:

int_0^1 x ln x , dx

Answer:

Use integration by parts:

Let u=ln x, dv=x,dx

int xln x dx = frac{x^2}{2}ln x – int frac{x^2}{2}cdot frac{1}{x} dx

= frac{x^2}{2}ln x – frac{x^2}{4}

Evaluate from 0 to 1:

= 0 – frac{1}{4}

Final Answer: boxed{-frac{1}{4}}

Question 5 Improper Integral

Determine whether the integral converges:

int_1^infty frac{1}{x^p}, dx

Find the values of p for which it converges.

Answer:

int_1^infty x^{-p} dx

Converges if p>1, diverges if p le 1.

Final Answer: Converges for boxed{p>1}

Question 6 Series

Determine whether the series converges:

sum_{n=1}^infty frac{n}{n^2+1}

Answer:

Compare with frac{n}{n^2} = frac{1}{n}.

Since the harmonic series diverges, the given series diverges by comparison.

Final Answer: Diverges

Question 7 Partial Derivatives

Given

f(x,y)=x^2y+3xy^2

Find f_{xy}.

Answer:

First f_x = 2xy + 3y^2

Then differentiate w.r.t. y:

f_{xy} = 2x + 6y

Final Answer: boxed{2x + 6y}

Question 8 Gradient & Directional Derivative

Find the directional derivative of

f(x,y)=x^2+y^2

at (1,2) in the direction of vector v=langle 3,4rangle.

Answer:

Gradient:

nabla f = langle 2x,2yrangle

At (1,2):

langle 2,4rangle

Unit vector in direction v:

frac{langle3,4rangle}{5}=leftlanglefrac35,frac45rightrangle

Directional derivative:

2cdotfrac35 + 4cdotfrac45 = frac{6+16}{5}=frac{22}{5}

Final Answer: boxed{frac{22}{5}}

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