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}}