
Math
You need to use the mean value thorem to verify the given inequality, such that: int_a^b f(x)dx = (ba)*f(c), c in (a,b) Replacing cos x for f(x) and pi/6 for a, pi/4 for b, yields:...

Math
`int_3^0(1+sqrt(9x^2))dx` Consider the graph of y=f(x)=`1+sqrt(9x^2)` `y=1+sqrt(9x^2)` `y1=sqrt(9x^2)` `(y1)^2=9x^2` `x^2+(y1)^2=3^2` This is the equation of circle of radius 3 centred at...

Math
`int_5^5(xsqrt(25x^2))dx` `=int_5^5xdxint_5^5sqrt(25x^2)dx` `=I_1I_2` I_1 can be be interpreted as area of two triangles;one above the xaxis and the other below axis.Since they are on the...

Math
`int_(1)^2 x dx` To interpret this in terms of area, graph the integrand. The integrand is the function f(x) =x. Then, shade the region bounded by the graph of f(x)=x and the xaxis in the...

Math
`int_0^10 x5dx` To interpret this in terms of area, graph the integrand. The integrand is the function f(x) = x  5. Then, shade the region bounded by f(x) = x5 and the xaxis in the...

Math
You need to use the mean value thorem to verify the given inequality, such that: `int_a^b f(x)dx = (ba)*f(c), c in (a,b)` Replacing `x^2  4x + 4` for `f(x)` and 0 for a, 4 for b, yields: `int_0^4...

Math
You need to check if` int_0^1 sqrt(1+x^2)dx <= int_0^1sqrt(1+x)dx` , using mean value theorem, such that: `int_a^b f(x)dx = (ba)f(c), ` where `c in (a,b)` `int_0^1 sqrt(1+x^2)dx <=...

Math
You need to use the midpoint rule to approximate the interval. First, you need to find `Delta x` , such that: `Delta x = (ba)/n` The problem provides b=8, a=0 and n = 4, such that: `Delta x =...

Math
You need to use the midpoint rule to approximate the interval. First, you need to find `Delta x,` such that: `Delta x = (ba)/n` The problem provides `b=pi/2` , a=0 and n = 4, such that: `Delta x =...

Math
You need to use the midpoint rule to approximate the interval. First, you need to find `Delta x` , such that: `Delta x = (ba)/n` The problem provides b=2, a=0 and n = 5, such that: `Delta x =...

Math
You need to evaluate the definite integral using the mid point rule, hence, first you need to evaluate `Delta x:` `Delta x = (ba)/n => Delta x = (51)/4 = 1` You need to denote each of the 4...

Math
You need to use the fundamental theorem of calculus, to prove the equality, such that: `int_a^b f(x)dx = F(b)  F(a)` You need to replace x for f(x), such that: `int_a^b xdx = x^2/2_a^b` `int_a^b...

Math
You need to evaluate the definite integral, such that: `int_a^b f(x) dx = F(b)  F(a)` `int_a^b x^2 dx = (x^3)/3_a^b` `int_a^b x^2 dx = (b^3)/3  (a^3)/3` `int_a^b x^2 dx = (b^3  a^3)/3` Hence,...

Math
You have to recall the definition of the Reiman Integral `int_a^bf(x)dx=lim_(n>oo)sum_(i=1)^nf(x(i))Deltax` `where Deltax =(ba)/n and x(i)= a +iDeltax` `x ` `a=2 and b = 6 ` `Deltax = (62)/n=...

Math
`int_1^10 (x4ln(x))dx` To express this definite integral as limit of Riemann's Sum, apply the formula: `int_a^b f(x) dx = lim_(n> oo)sum_(i=1)^oo f(x_i)Delta x` where `Delta x = (ba)/n` `x_i...

Math
`int_1^2(1x)dx` To interpret the integral in terms of area , graph the integrand. The integrand is the function `f(x)=1x` Graph the function in the interval (1,2). Refer the attached graph. The...

Math
`int_0^9 (1/3x2)dx` To interpret this integral in terms of area, graph the integrand. The integrand is the function `f(x)=1/3x2` . Then, shade the region bounded by `f(x)=1/3x2` and the...

Math
`v(t)=2t1/(1+t^2)` position of the particle s(t) is given by, `s(t)=intv(t)dt` `s(t)=int(2t1/(1+t^2))dt` `s(t)=2(t^2/2)arctan(t)+C` , C is constant `s(t)=t^2arctan(t)+C` Now let's find C ,...

Math
`a(t) = sin(t) + 3cos(t)` `v(t) = cos(t) + 3sin(t) + a` `Now, v(0) = 2` `Thus, 2 = cos(0)+3sin(0)+a` `or, 2 = 1 + 0 + a` `or., a = 3` `Now, v(t) = cos(t) + 3sin(t) + 3` `Thus, s(t) = sin(t) ...

Math
You need to evaluate the asimptotes of the function, such that: `lim_(x>pi/2,x>pi/2)(4x  tan x) = 2pi  tan(pi/2) = 2pi + tan (pi/2) = oo` `lim_(x>pi/2,x<pi/2)(4x  tan x) = 2pi...

Math
`y=(x2)e^(x)` (I) Asymptotes To determine its horizontal asymptotes, take the limit of this function as x approaches positive and negative infinity. `lim_(x>oo) (x2)e^(x) =oo`...

Math
`y=f(x)=x+ln(x^2+1)` a) Asymptotes The function has no undefined points , so it has no vertical asymptotes. For Horozontal asymptotes , check if at `x>+oo` the function behaves as a line...

Math
`f'(x) = cos(x)  (1x^2)^(1/2)` `f(x) = sin(x)  sin^1(x) + c` ``

Math
`f'(x) = 2e^x + sec(x)tan(x)` `f(x) = 2e^x + sec(x)` ``

Math
`f'(x)=sqrt(x^3)+root(3)(x^2)` `f(x)=intf'(x)dx` `f(x)=int(sqrt(x^3)+root(3)(x^2))dx` apply the sum rule, `f(x)=intsqrt(x^3)dx+introot(3)(x^2)dx` `f(x)=x^(3/2+1)/(3/2+1)+x^(2/3+1)/(2/3+1)+C` , C is...

Math
You need to evaluate the function f using the provided information, hence, you need to apply the antiderivative, such that: `int (f'(x) sinhx + 2cosh x)dx = int (f'(x)(e^x  e^(x))/2) + (e^x +...

Math
`f'(t)=2t3sin(t)` `f(t)=int(2t3sin(t))dt` `f(t)=2(t^2/2)3(cos(t))+C` ,C is constant `f(t)=t^2+3cos(t)+C` Now , evaluate C , given f(0)=5 `f(0)=5=0^2+3cos(0)+C` `5=3+C` `C=2`...

Math
You need to evaluate f(u) using the antiderivative of the function f'(u), such that: `int f'(u) du = f(u) + c` `int (u^2 + sqrt u)/u du = int (u^2)/u du + int (sqrt u)/u du` `int (u^2 + sqrt u)/u...

Math
`f''(x) = 1  6x + 48x^2` `f'(x) = x  3x^2 + 16x^3 + a` `Now, f'(0) = 2` `i.e. 2 = a` `Thus, f'(x) = x  3x^2 + 16x^3 + 2` `f(x) = (x^2/2)  x^3 + 4x^4 + 2x + b` `Now, f(0) = 1` `Thus, b = 1`...

Math
`f''(x) = 2x^3 + 3x^2  4x + 5` `or, f'(x) = {(1/2)x^4} + x^3  2x^2 + 5x + a` `or, f(x) = {(1/10)*x^5} + (x^4/4)  {(2/3)x^3} + {(5/2)x^2} + ax + b` `Now, f(0) = 2` `i.e. 2 = b` `Also, f(1) = 0`...

Math
`lim_(x>oo)(x^2x^3)e^(2x)` `=lim_(x>oo)x^3(1/x1)e^(2x)` Apply the Algebraic limit theorem, `=lim_(x>oo)x^3(lim_(x>oo)1/x1)lim_(x>oo)e^(2x)` plug in the value of x...

Math
You need to evaluate the limit, hence, you need to replace` pi` for x in expression under limit: `lim_(x>pi)(xpi)csc x = (pipi)/(sin pi) = 0/0` Since the limit is indeterminate `0/0` , you...

Math
`lim_(x>1^+)(x/(x1)1/ln(x))` `=lim_(x>1^+)(xln(x)1(x1))/((x1)ln(x))` `=lim_(x>1^+)(xln(x)x+1)/((x1)ln(x))` Apply L'Hospital's rule, Test L'Hospital condition :0/0...

Math
`lim_(x>(pi/2)^)(tan(x))^cos(x)` `=lim_(x>(pi/2)^)e^(cos(x)lntan(x))` applying the limit chain rule, `lim_(x>(pi/2)^)cos(x)lntan(x)` `=lim_(x>(pi/2)^)(lntan(x)/(1/cos(x)))`...

Math
You need to evaluate the horizontal asymptotes to the graph of function, such that: `lim_(x>+oo) 22xx^3 = oo` `lim_(x>oo) 22xx^3 = oo` Notice that the graph has no vertical or...

Math
`y=x^36x^215x+4` a) Asymptotes Polynomial function of degree 1 or higher can't have any asymptotes. b) Maxima/Minima `y'=3x^212x15` `y'=3(x^24x5)` `y'=3(x5)(x+1)` Now we can find critical...

Math
The function has no vertical, horizontal or slant asymptotes. You need to determine the extrema of the function, hence, you need to determine the zeroes of the first derivative: `f'(x) =...

Math
`y=x/(1x^2)` a) Asymptotes Vertical asymptotes are the zeros of the denominator of the function. `1x^2=0rArr(1+x)(1x)=0` `x=1 , x=1` Vertical asymptotes are x=1 and x=1 Degree of numerator=1...

Math
`y=1/(x(x3)^2)` a) Asymptotes Vertical asymptotes are the zeros of the denominator. `x(x3)^2=0rArr x=0,x=3` So, x=0 and x=3 are the vertical asymptotes. Degree of numerator=0 Degree of...

Math
`y=1/x^21/(x2)^2` To determine the asymptotes, express the function as a single fraction. Since the LCD of the two fractions is x^2(x2)^2, then, the function becomes: `y = 1/x^2 *...

Math
`y=x^2/(x+8)` a) Asymptotes Vertical asymptotes are the zeros of the denominator `x+8=0rArrx=8` Vertical asymptote is x=8 Degree of numerator=2 Degree of denominator=1 Degree of...

Math
`y=sqrt(1x)+sqrt(1+x)` Domain of function: `1<=x<=1` a) Asymptotes a) Asymptotes The function has no undefined points, so there are no vertical asymptotes. For horizontal asymptotes , check...

Math
`y=xsqrt(2+x)` a) Asymptotes Domain of function is x `>=` 2 Since the function has no undefined points , so it has no vertical asymptotes. Horizontal Asymptotes: Let's find the limits of the...

Math
`y=root(3)(x^2+1)` a) Asymptotes Since the function has no undefined point, so it has no vertical asymptote. For horizontal asymptotes check if at x`>+oo` , the function behaves as a line...

Math
You need to evaluate the local absolute extrema of the function, hence, you need to find the zeroes of the equation f'(x) = 0. You need to evaluate the derivative such that: `f'(x) = 3x^2  12x +...

Math
You need to evaluate the local absolute extrema of the function, hence, you need to find the zeroes of the equation f'(x) = 0. You need to evaluate the derivative using product rule, such that:...

Math
You need to evaluate the local absolute extrema of the function, hence, you need to find the zeroes of the equation f'(x) = 0. You need to evaluate the derivative using the quotient rule: `f'(x) =...

Math
`f(x)=sqrt(x^2+x+1)` Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers that are in the interval and evaluate the...

Math
You need to evaluate the extreme values of the function on the interval [pi,pi], hence, you need to evaluate the zeroes of the function f'(x). You need to find the derivative of the function:...

Math
`f(x)=x^2e^x` Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers that are in the interval and evaluate the function at...