1) If f(x)= (x²- 5x +6)/ (x²- 8x +12) show that f(2) is not defined and also find f(-5). 8/11
2) If f(x)=eᵃˣ⁺ᵇ,. show that eᵇ. f(x + y)= f(x).f(y).
3) If f(x)= |x| - |-x|. Where |x| is the greatest integer not exceeding x, find the value of f(3.5) and f(-3.5). 0.5,7.5
4) Find the domain of definition of the function 5/√{(x +1)(x -3)}. - ∞<x <-1 and 3<x < ∞
5) If f(x)= |x| - 2x, find f(-1), f(1). 3,-1
6) If f(x)= (ax - b)/(bx - a), show that f(a). f(1/a) - f(b) f(1/b)= 0.
7) Show that the function f(x, y)= (5x²- 7y²)/3y² is a homogeneous function.
8) If f(x)= (eˣ -1)/(eˣ +1) and f(a)= (1+ f(x))/(1- f(x)), then show that f(x + a)= f(x). f(a)
9) If f(x)= (1+ eˣ)/(1- eˣ) , then show that f(x) is an odd function.
10) If f(x)=x²- x, then show that f(h +1)= f(-h)
11) If f(x)= (ax + b)/(bx + a) show that f(x). f(1/x)= 1
12) If y= f(x)= (x + 1)/(x + 2), find f(y) and f{f(1/x)}. (2x+3)/(3x +5), (2+ 3x)/(3+ 5x)
13) If f(x)= (x -1)/(x +1), then show that {f(a) - f(b)}/(1+ f(a)f(b))= (a - b)/(1+ ab)
14) If f(x)= (2x +1)/(2x²+1) , g(x)= 2f(2x), then find g(24). 22/51
15) If f(x +3)= 3x²- 2x +5, find f(x -1). 3x²- 26x +61
16) Find {f(x+ h) - f(x)} when f(x)= (1- x)/(1+ x).
17) A function is defined as follows
f(x) = x when x > 0, = - x when x=0 obtain lim ₕ→₀₊ (f(h) - f(0))/h and lim ₕ→₀₋ f(h) - f(0)/h
What can you say about the derivative of f(x) at x=0. The function does not exist.
18) Find the domain of definition of the following function:
a) (x +2)/√(x²- x -2). ∞ ≤ x < -1, 2≤ x < ∞
b) (4x -5)/√(x²- 7x +12). -∞ ≤ x < 3, 4≤ x < ∞
c) log((x²- 5x +6). x> 3, x> 2 or -∞ ≤ x < 2, 3≤ x < ∞
d) (x²- -5x +6)/(x²- 8x +12). Domain of the function is all values of x except 6.
19) Find the range of the function 2x/(4 + x²) where x is real -1/2 ≤ y < 0, 0≤ y ≤ 1/2
20) If y= f(x)= (ax + b)/(CX + a), then show that f(y)= x.
21) If x is real, Find the range of the function x/(1+ x²). -1/2≤ y ≤1/2
22) If f(x)= (ax - b)/(bx - a), show that f(a) - f(b)f(1/b)= 0
LIMITS
1) im ₓ→₁ (x²- 2x +3)/(x +4). 2/5
2) lim ₓ→₀ (x²+ x -12)/(x -3). 7
3) lim ₓ→₃ (x²- 9)/(x -3). 6
4) lim ₓ→₂ (x²+ x -6)/(x²- x -2). 5/3
5) lim ₓ→ₐ (√x - √a)/(x - a). 1/2√a
6) lim ₓ→ₐ (x¹⁾³ - a¹⁾³)/(x - a). 1/3³√a²
7) lim ₓ→∞(1- √x)/(1+ √x). -1
8) lim ₓ→₋₄ [1/(x +4) + 1/(x²-4)]. Does not exist
9) lim ₓ→₂ (2x²- 7x +6)/(3x² -7x +2). 1/5
10) lim ₓ→ₐ {√(2x +a) - √(x + 2a)}/(x - a). 1/2√3 a
11) lim ₓ→₂ (x²+ x -6)/(x²- x -2). 5/3
12) lim ₓ→₀ {√(a+ x) - √a}/x. 1/2√a
13) If lim ₓ→₂ (ax²- b)/(x -2)= 4, find the values of a and b. 1,4
14) lim ₓ→₁(x²+ 5x -6)/(x²- 3x +2). -7
15) lim ₓ→₂ {√(x +7) -3}/(x -2). 1/6
16) lim ₓ→₀ {√(1+ 2x) - √(1- 3x)}/x. 5/2
17) lim ₓ→₃ (x²+ x - 12)/(x -3). 7
18) lim ₓ→₀ {√(x + h - √x}/h. 1/2√x
19) lim ₕ→₀ {f(1+ h) - f(1)}/h when f(x)= 1/x
20) lim ₓ→₃ (x -3)/{√(x -2) - √(4- x)}. 1
21) lim ₓ→₂(x²- 5x +6)/(x²- 3x +2). -1
22) lim ₓ→ ₋ ₁ (2x² - x -3)/(x²- 2x -3). 5/4
23) lim ₓ→₀ {√(1+ x) -.√(1- x)}/x. 1
24) lim ₓ→₁ (x²- 1)/{√(3x -1) -.√(5x -1)}. -4
25) lim ₓ→₁(x² -3x +2)/(x²- 4x +3). 1/2
26) lim ₓ→₋₁(2x²- x -3)/(x²- 2x -3). -1
27) lim ₓ→₀ (eᵃˣ - eᵇˣ)/x. (a- b)
28) lim ₓ→ₐ (3x⁴+ 2x²+ 1)/(x⁴+ 2x²+1). 3
29) lim ₓ→₁ (x²+ 4x -5)/(x -1). 6
30) Lim ₓ→₃ (x -3)/{√(x -2) - √(4- x)}. 1
31) lim ₓ→∞ (4x²- 3x +2)/(5x⁴+ 2x²+3). 0
32) lim ₓ→₀ x/{√(1+ x) - √(1- x)}.. 1
33) lim ₓ→∞ (3x³ + 2x -1)/(4x³+ 3x²-2) . 3/4
34) lim ₕ→₀ {f(2+ h) - f(2)}/h where f(x)= 2x²- x +1. 7
35) lim ₓ→₂₅ {(√x -5)(x +1)}/(x²- 24x -25). 1/10
36) lim ₓ→∞ (3h⁴- 2h² +1)/(h⁴- 2h²+3). 3
37) lim ᵧ→₀ (1/y) {√(1+ 2y) - √(1- 2y)}. 2
38) lim ₓ→₀ {(√1+ x²) - √(1+ x)}/{√(1+ x³) - √(1+ x)}. 1
39) lim ₓ→₅ (x³-125)/(x⁴- 625). 3/20
40) lim ₓ→∞ (x² + 3x +2)/(x³ + x -4). 0
41) lim ₓ→₀ {√(x²+ a) - √(a - x²)}/x². 1/√a
41) lim ₓ→₃ (x² + x -12)/(x²+2x - 15). 7/8
42) lim ₓ→₀ {√(1+2x) - √(1- 3x)}/x. 5/2
43) lim ₓ→∞ (5x²- 3x +7)/(3x²+ x +4). 5/3
44) lim ₓ→₋₄ [1/(x +4) + 8/(x²-4)]. Does not exist
45) lim ₓ→₂ (2x²- 7x +6)/(3x²- 7x +2). 1/5
46) lim ₓ→₀ {f(x +h) - f(x)}/h, where f(x)= 1/√x. (x > 0). -1/2√x³
47) lim ₓ→∞ (15x⁷ + 12x +17)/(5x⁷+ 9x²+12). 3
48) lim ₓ→₀ {√(a+ x) - √a}/2x. 1/4√a
49) lim ₓ→∞ (3x²- 4x +6)/(x²+ 6x -7). 3
50) limₓ→₀ (e³ˣ - e²ˣ + 2x)/x. 3
51) lim ₓ→₀ (14ˣ - 7ˣ - 2ˣ +1)/x². Logₑ7 logₑ2
52) lim ₕ→₀ {f(2+ h) - f(2)}/h, where f(x)= 2x²- 7x +1. 1
53) lim ₓ→₀ ∈ ∞ lim ₓ→ₐ lim ₙ→∞
54) lim ₓ→₃ {x - √(x-a)(x -b)/(x²+2x - 15). (b + a)/2
55) lim ₓ→₃ (√x - √3)/(x²- 9). 1/12√3
56) lim ₓ→∞{1/(1+ n) + 1/(2+ n) + 1/3- n) + ....+ 1/2n}. 0
CONTINUITY
1) f(x)= (x²- 9)/(x -3). When x≠ 3. State the value of f(3) so that f(x) is continuous at x=3.
2) Find f(2) so that f(x)= (x²- 4)/(x -2) may be continuous at x=2.
3) For what value of f(3), f(x)= (x²-9)/(x -3) will be continuous at x= 3 ?
4) Draw the graph of f(x)= x²/x and g(x)= x. Rough sketches only are to be given. From the graphs so drawn state which of the two functions is not continuous? What is the point of discontinuty ? Indicate the point of discontinuty of the function (2x²+ 6x -5)/(12x²+ x -20).
5) Sketch the graph of f(x)= |x|/x . From the graph, examine continuity of f(x) at x=0.
6) Sketch the graph of f(x)= |x|. From the graph, examine continuity of f(x) at x=0.
7) Sketch the graph of f(x)= 1 for x ≥ 0
= -1 for x ≤ 0
From the graph discuss whether lim ₓ→₀ f(x) exist or not
8) A function f(x) is defined as follows
f(x)= 3+ 2x for -3/2≤ x < 0
= 3- 2x for 0≤ x < 3/2
= -3 - 2x for x ≥ 3/2
Show that f(x) is continuous at x=0 and discontinuous at x= 3/2.
9) Sketch the graph of
f(x)= 2x +1 when x≥ 1
= 2x -1 when x < 1
From the graph examine whether f(x) is continuous at x= 1 or not
10) Draw the graph of the following function
f(x)= 1 when x > 0
= 0 when x = 0
= -1 when x < 0
Examine the continuity of f(x) at x=0 from the graph.
11) Sketch the graph of
f(x)= 3x +1 when x ≥ 1
= 3x -1 when x < 1
From the graph examine whether f(x) is continuous at x= 1 or not
12) f(x)= (x²-9)/(x -3), when x≠ 3. State value of f(3) so that f(x) is continuous at x= 3.
13) Sketch the graph of the function
f(x)= - x when x ≤ 0
= x when 0< x.
From the graph examine the continuity of f(x) at x= 0. Con
14) Sketch the graph of the function
f(x)= 3+ 2x when x ≤ 0
3 -2x when x> 0
From the graph examine the continuity of f(x) at x=0. C
15) Sketch the graph of the function defined by
f(x)= x -1 when x> 0
= 1/2 when x =0
= x +1 when x< 0
From the graph examine continuity of f(x) at x = 0. D
16) Draw the graph of the following function
f(x)= 2x -1. 0≤ x ≤ 4
= 2- x² -4< x < 0
State from the graph whether f(x) is continuous at x = 0.
17) Draw a rough sketch of the function f(x)= x/|x| and discuss its continuity at x=0. D
18) Examine the continuity of the function defined by
f(x)= x -1 when x> 0
=1/2 when x= 0
= x +1 when x < 0
19) A function f(x) is defined as follows:
f(x)= |x -3|/(x -3), if x ≠3
=. 1 if x= 3
Discuss the continuity of f(x) at x = 3. D
20) Given f(x)= (x²-4)/(x -2), if x≠ 2. Find the value of f(2) and show that f(x) is continuous at x = 2.
21) Discuss continuity of f(x) at x = -2, where
f(x)= {x + (x +2)/|x +2|, if x≠ -2
= -1, x = -2. C
22) Examine the continuity of the function
f(x)= 2- 3x when x > 0
= 2 when x = 0
= 2+ 3x when x< 0 at x = 0. C
23) If f(x)= (6- 4x)/(1+ 2x + 2x²), find f(0). Is the function continuous at x= 0 ? Y
DIFFERENTIATION
Find the first principle the derivative of
1) 1/x³. -2/x³
2) x²- 2x. 2x - 2
3) √x at x = 4. 1/4
4) 2x³+ 3. 6x²
5) 3x³ + 7. 9x²
6) 5x²+ 2. 10x
7) x³+ 4 at x = 1. 3
8) Evaluate lim ₕ→₀ {f(x+ h) - f(x)}/h where f(x)= 2x²+ 3x - 4. 4x +3
9) (x -1)³ at x=1. 0
10) x³. 3x²
11) (3- 5x)¹⁾². (-15/2)+ √(3- 5x)
12) 2ˣ logx. 2ˣ(1+ x logx log2)/x.
13)
DIFFERENTIATE
1) y= ₑax²+ bx + c. (2ax + b)ₑax²+ bx + c.
2) x/(eˣ -1). (eˣ(1- x) -1)/(eˣ - 1)².
3) 2ˣ. x⁵ . 2ˣx⁴(5+ x log 2)
4) (x² - 3x -5)¹⁾². (3(2x -3)√(x²- 3x -5))/2
5) (2- 5x)¹⁾². (15/2) √(2- 5x).
6) x⁵⁾² logx. x³⁾²(1+ (5/2) logx)
7) If y= x/√(1- x²) then show that (1- x²) dy/dx = y/x.
8) x= at², y= 2at.. 1/t
9) x²+ y²= 2a². -x/y
10) 3⁴ˣ + 3/³√x. 3⁴ˣ. 4 logè - 1/³√x⁴
11) (5- 4x)/(5+ 4x). -40/(5+ 4x)²
12) log(x + √(x²+ a²)). 1/√(x²+ a²)
13) If xᵐ yⁿ = (x + y)ᵐ⁺ⁿ show that dy/dx = y/x.
14) x². 5³ˣ. x. 3³ˣ(2+ 3x log5)
15) (x²- 2x -3)/(x - 1). (x²- 2x +5)/(x - 1)²
16) xʸ. yˣ= 1. -(y + x logy)y/(y logx + x)x
17) (x²+1)/(x -1). (x²- 2x - )/(x - 1)²
18) xˣ. (1+ log x)xˣ
19) xʸ + y = 1. yxʸ⁻¹/(1+ xʸ log x)
20) 10ˣ. x¹⁰. 10ˣ. x⁹(x log 10 + 10)
21) x²/a² + y²/b² = 1. - b²x/a²y
22) xʸ + xy = 8. -y(1+ xʸ⁻¹)/x(xʸ⁻¹ log x +1)
23) xˣ + x². xˣ + xˣ log x + 2x
24) √(x²+ a²). x/√(x²+ a²)
25) (x²+ 1)eˣ. eˣ(x +1)²
26) (1+ x)ˣ. x(1+ x)ˣ⁻¹ + (1+ x)ˣ log(1+ x)
27) x³+ 3x²y + y³ = a³. -(x²+ 2xy)/(x²+ y²)
28) 3x²- x²y + 2y³= 0. (2xy - 6x)/(6y²- x²)
29) 7²ˣ+ 2ˣ. 7²ˣ2 log 7 + 2ˣ log 2.
30) x= y log(x²y²). (x - 2y)/(2(log xy +1).
31) y= xˣ. (1+ logx)xˣ
32) x³+ y³= 3axy. (ay - x²)/y²- ax)
33) x²/a²+ 2xy/h + y²/b²= 1. -(a²y + hx)b²/(b²x + hy)a².
34) x= ct, y= c/t. - 1/t²
35) eˣʸ = 4(1+ xy) and eˣʸ ≠ 4 then show dy/dx = -y/x.
36) xʸ = yˣ. (x logy - y)y/(y logx - x)x.
37) y=√{(1+ x)/(1- x)}. 1/{(1- x)√(1- x²)}
38) x= t/(1+ t), y= t/(1- t). {(1+ t)/(1- t)}²
39) x⁴ₑ3x². 2x³(2+ 3x²)ₑ3x²
40) 3x²+ 2xy - y²= 4. -(y + 3x)/(x - y)
41) aˣ + xʸ = 4. -(aˣ log a + yxʸ⁻¹)/(xʸ logx)
42) xˣ +2. (1+ logx)xˣ + 2ˣ log
43) x= log(xy). (x - y)/x(1+ log(xy))
44) x= ct³, y= c/t². -1/t⁶
45) s= t¹⁻ᵗ + t⅖ find ds/dt. t¹⁻ᵗ{(1- t)/t logt)} + 2t
46) (eˣ +1)y= eˣ -1. e(1- y)/(eˣ +1)
47) y= (xˣ)ˣ. ₓxˣ⁺¹(1+ 2 logx)
48) yˣ = eˣ. (1/e - logy)y/e
49) x= √(1+ t); y= √(1- t) at t= 1/2. - √{(1+ t)/(1- t)}
50) xy = eˣ⁻ʸ show that dy/dx = logx/(1+ logx)²
51) log{√(x - a) + √(x - b)}. 1/2√{(x - a)(x - b)}
52) x⁴+ x²y²+ y⁴= 0. -x(2x² + y²)/y(x²+ 2y²)
53) y= (1+ 2x)ˣ . (1+ 2x)ˣ .{2x/(1+ 2x) + log(1+ 2x}
54) y= log(ax²+ bx + c). (2ax + b)/(ax²+ bx + c).
55) x= 1/(1+ t), y= 1/(1- t). {(1+ t)/(1- t)}²
56) y= xˣ+ log(3x²+ 4x+ 5). xˣ(1+ logx). (6x+4)/(3x²+ 4x +5)
57) x= 3at/(1+ t³), y= 3at²/(1+ t³). (2t - t⁴)/(1- 2t³)
58) y= x²/(a²- x²) at x=1. 2a²/(a²- 1)²
59) y= e²ᵐˣ + e⁻²ᵐˣ show that d²y/dx²- 4m²y =0
60) ₓxˣ. ₓxˣ{xʸ/x + log xˣ(1+ logx)}
61) x²ᵖ yᑫ = (x + y)²ᵖ⁺ᑫ. y/x
62) x= 2at/(1+ t)², y= (1- t²)/(1+ t²). -2t/a(1- t²)
63) xˣ + ₑax²+ bx + c. xˣ(1+ logx) + ₑax²+ bx + c(2ax + b)
64) If y= (x + √(1+ x²))ᵐ, show that (1+ x²)y₂ + xy₁ - m²y=0
65) If y= (x -2)/(x +2), show that 2x dy/dx = 1- y².
66) If yˣ = eʸ⁻ˣ, show that dy/dx = (logₑy)²/logy.
67) Given x= t + 1/t and y= t - 1/t, find value of d²y/dx⅖ at the point t=20.
68) If x³- 2x²y²+ 5x + y -5=0, then find d²y/dx² at x= 1 , y= 1
69) If f(x)= {(a + x)/(b + x)}ˣ + 2x, show that f'(0)= 2 + log(a/b).
70) If x⁴ + x³y³+ y⁴= 0
71) f(x)= {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ show that f'0)= (2 log(a/b) + (b²- a²)/ab)(a/b)ᵃ⁺ᵇ
EULER'S THEOREM
1) State Euler's theorem for homogeneous function of 2 variables of degree 'n'.
MAXIMUM AND MINIMUM
1) Find maximum and minimum values of the function y= x³- 3x -1. 3,-1
2) Find the maximum or minimum point of the function y= 5- x - x²
3) Show that the maximum value of the function 2x + 1/2x is less then its minimum value.
4) Examine x³- 9x²+ 24x -12 of maximum and minimum values of x³- 9x²+ 24x -12.
5) Show that the function f(x)= 12 -24x - 15x²- 2x³ has a maximum at x= -1, minimum at x= -4 and point of inflexion at x= 5/2.
6) Show that the the maximum value of x³+ 1/x³ is less than its minimum value.
6) Find the maximum and minimum values of y= x/{(x -1)(x -4)}. -2
7) Determine whether the following 2x³/3 - 6x²+ 20x - 5 has a maximum or a minimum.
8) Show that maximum value of f(x)= x +1/x is less than its minimum value.
9) Examine whether the curve y -3 = 6(x - 2)⁵ has a point of inflexion at (2,3).
10) Show that maximum value of 2x + 1/x is less than its minimum value.
11) Find the point of inflexion of the curve y+ 5= x³- 3x²+ 9x.
12) Find the maximum and minimum value of the function 2x³- 15x²+ 36x. 28,27
13) Divide the number 20 in two parts in such a way that their product will be maximum. 10,10
14) If x+y= 2, show that the maximum value of (4/x + 36/y is less than its minimum value.
APPLICATION OF DERIVATIVES (COMMERCE)
1) A firm produces x tons of a valuable metal per month at a total cost c given by:
C= Rs (x³/3 - 5x² + 75x +10).
Find at what level of output the marginal cost attains its minimum. 5
2) A firm produces x units of output per week at a total cost of Rs (x³/2 - x² + 5x +3).
Find at what level of output the marginal cost and the average variable cost attains their respective minima. 2/3
3) A radio manufacturer finds that he can sell x radios per week at Rs p each where p= 2(100 - x/4). His cost of production of x radios per week is Rs (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week. 40, Rs1600
4) A radio manufacturerproduces x sets per week at total cost of Rs x²+ 78x + 2500. He is monopolist and the demand function for his product is x= (600- p)/3 when the price is Rs p per set. Show that the maximum net revenue (profit) is obtained when 29 sets are produced per week. What is the monopoly price? Rs405
5) A manufacturer can sell x items per month at a price p= 300 -2x rupees. Manufacturer's cost of production y rupees of x items is given by y= 2x + 1000. Find the number of items to be produced per month to yield the maximum profit. 75
6) The cost of manufacturing a certain article is given by the formula C= 5+ 48/(x + 3x²) where x is the number of articles manufactured. Find the minimum value of C. 41
7) The total cost function C for producing x units of an article per day is given by C= Rs (400 -16x + x²). Find the average cost function and the level of output at which their function is minimum. 20
8) If the cost function for x units is given by C= Rs (400- 15x - x²) obtain the
a) average cost
b) average variable cost. 400/x - 16 - x, (-6- x)
9) The total cost C, of making x units of a product is C= axⁿ + b, where a,b,n are positive constants. Find marginal cost and marginal average cost. naxⁿ⁻¹, axⁿ⁻¹ + b/x, a(n -1)xⁿ⁻² - b/x²
INTEGRATION
1) ∫ x/(x +1) dx. x - log(x +1)
2) ∫ √x(x½+ 3x +4)dx. (2/7) √x⁷ + (6/7) √x⁵+ (8/3) √x
3) ∫ (x²+ 1)²/x³ dx. x²/2 + 2 logx - (1/2x²)
4) ∫ 3dx/(x²-1). (3/2) Log{x -1)/(x +1)
5) ∫ √{(x +1)/(x -1)} dx. √(x²-1) + log|x + √(x²-1)|
6) ∫ x²/³√(3x +5) dx. (1/8) ³√(3x +5)⁸ - 2 ³√(3x +5)⁵ + (25/2) ³√(3x +5)²
7) ∫ logx/(x +1)² dx. - log(x +1) + log|x/(x +1|
8) ∫ x⁴/(³√(2x⁵+3) dx. (3/20)³√(2x⁵+ 3)²
9) ∫ (x -2)eˣ/(x -1)² dx. eˣ/(x -1)
10) ∫ dx/(eˣ+1). eˣ/(eˣ +1)
11) ∫ x √(x²-1) dx. (1/4) (x⁴- 2x²+1)
12) ∫ (1+ 2/√x +3) dx. 4√x(√x + 1)
13) ∫ x³/(x -1) dx. . (2x³+ 3x²+ 6x -11 + 6 log(x -1)/6
14) ∫ (4x -3)/x² dx. 32x²- 144x + 27/x + 108 logx
15) ∫ (x -2) dx/³√(x²- 4x -5). (3/4) ³√(x²- 4x -5)²
16) ∫ dt/(2t²+ 3t +1). (1/2) Log|(2t+1)/(t+1)|
17) ∫ (x +1)²/√x dx. 2∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫
) Find the slope of the curve at the point t=2 when x= t²- 3., y= 2t +1. 1/2
) Find the gradient of the curve log(xy)= x²+ y² at the point (1,1). -1
) State Rolle's Theorem.
)
DIFFERENTIATION
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