Tangent and Normal
1) Write down the equation of the tangent at a point (4,3) on the ellipse 9x²+ 16y²= 288.
2) Write down the equation of the tangent of the parabola y²= 4x at (1,2).
3) write down the value of slope of the tangent to the parabola y² = 8(x - 6) at the point (8,-4).
4) Determine the points on the curve y= x+ 1/x, where the tangent is parallel to x-axis.
5) Find the gradient of the tangent to the parabola y² = 4x at the point (1,2).
6) Show that the equation of the normal to the hyperbola x = a sec t, y= b tan t at the point (a sec t, b tan t) is . ax cos t + by cot t= a²+ b².
7) At a point P(x,y) on the parabola y= x²/4 + 1, the tangent makes an angle 45° with the positive direction of the x-axis. Find the coordinates of the point P.
8) Find the condition that the straight line lx + my = n touches the ellipse x²/a² + y²/b² = 1.
9) Write down the equation of the tangent and normal of the parabola y²= 4ax at (0,0).
10) Obtain the equation the normal to the hyperbola x²/a² - y²/b² = 1 at (a sec t, b tan t).
11) If the straight line lx + my = 1 is a normal to the parabola y²= 4ax, then show that al³+ 2alm² = m².
12) Find the coordinates of the point on the curve y= 1+ 2x- 3x² at which the tangent make an angle of 45° with the positive direction of the x-axis.
13) Find the equation of the tangent to the hyperbola x²/a² - y²/b² = 1 at (a sec t, b tan t).
14) find the equations of the tangent and normal to the ellipse 4x² + 9y² = 72 at the point (3,2).
15) Find the equation of the tangent to the parabola y² = 4x + 5, which is parallel to the straight line y= 2x+ 7.
16) Find the equation of the normal to the curve y= x² - x at the point (3,6). Show that this normal touches the parabola x² + 660y = 0.
17) If the straight line lx + my = n is normal to the hyperbola x²/a² - y²/b² = 1, then show that a²/l² - b²/m² = (a² + b²)²/n².
18) If the straight line x cos t+ y sin t= p touches the ellipse x²/a² + y²/b² = 1, then prove that a² cos²t + b² sin²t = p².
19) Prove that the straight line x+ y= 2+ √2 touches the circle x²+ y² - 2x - 2y+ 1= 0. Find the point of contact.
20) If the straight line lx + my + n= 0 touches the parabola y²= 4ax then prove that am² = nl.
21) tangent are drawn from origin to the curve y= sinx. Prove that their points of contact lie on x²y² = x² - y².
22) Find the equation of the tangent and the normal to the curve y= x³ - 3x at the point (2,2).
23) Find the equation of the tangent and normal of the curve y(x - 2)(x -3) - x +7= 0 at the point of intersection with the x-axis.
24) Find the equation of the common tangent to the parabola y² = 4ax and x²= 4by.
25) Show that the normal at any point on the curve x= a(cos t+ t sin t), y= a(sin t - t cos t) is a constant distance from the origin.
Continuity
1) A function f(x) is defined as:
= x² when x< 1
f(x)= 2.5 when x= 1
= x²+2 when x> 1
Is f(x) continuous at x= 1?
2) A function f(x) is defined as:
= 3+ 2x when -3/2≤x< 0
f(x)= 3- 2x when 0≤ x< 3/2
= -3x -2x when x ≥ 3/2
Show that f(x) continuous at x= 0 and discontinuous at x=3/2.
3) A function f(x) is defined as:
=(Sin 3x)/2x when x≠ 0
f(x)= 2/3 when x= 0
Is f(x) continuous at x= 0 ?
4) Find the points of discontinuity of the function (x²+2x+5)/(x²-7x+12).
5) A function f(x) is defined as:
f(x)=(x⁴+4x³+2x)/sin x when x≠0
And f(0)=0
Show that f(x) continuous at x= 0
6) A function f(x) is defined as:
f(x)= x sin(1/x) when x≠ 0
= 0 when x = 0
Show that f(x) continuous at x= 0
7) The function f(x) is defined as:
=x²-2x+3 when x< 1
f(x)= 2 when x= 1
=2x²-5x+5 when x> 1
Is f(x) continuous at x= 1 ?
8) f(x)= (x²-1)/(x³-1) is undefined at x= 1. What should be the value of f(x) at x= 1 such that f(x) may be continuous at x= 1 ?
9) For what value of f(4), the function f(x) = (x²-16)/(x-4) is continuous at x= 4 ?
10) Examine the continuity of the function f(x) at x= 1.
f(x)= |x-1|/(x-1) when x≠1
= 0 when x = 1
Is f(x) continuous at x= 1?
11) Test whether the following function is continuous at x= 0 :
f(x)= |x|/x when x≠ 0
= 1 when x = 0
12) A function f(x) is defined as:
= x+1 when x≤ 1
f(x)= 3- ax² when x > 1
Find the value of a for which f(x) will be continuous at x= 1 ?
13) A function f(x) is defined as:
f(x)=(x³+x²-16x+20)/(x-2)² at x≠2
= K at x= 2
If f(x) continuous for all values of x then find the value of K.
14) Determine the values of a so that the following function is continuous at x= 1
f(x)= ax+3, when x ≥ 1
= x²+a² when x < 1
15) Two function f(x) and g(x) are defined as:
f(x)= x²+4 when x ≤ 2
= x+6 when x > 2
And g(x)= 2x when x ≤ 2
= 4 when x > 2
Show that f(x). g(x) is continuous at x = 2.
16) The function f(x)={log(1+ax) - log(1- bx)}/x is not defined at x= 0. Find the value of f(0), so that f(x) is continuous at x= 0.
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Answers
1) no 3) no 4) 4,3 7) yes 8) 2/3 9) 8
10) continuous at x= 1
11) discontinuous at x= 0 12) 1
13) 7 14) 2, -1 16) a+ b
DIFFERENTIATION
1) logₑ x
2) log₁₀x
3) x ⁻³⁾²
4) x + 1/x
5) x⁴ +6
6) sin 4x
7) sin(x²)
8) ₑ√x
9) log(sin x)
10) cos ⁻¹(ₑ√tanx)
11) a |sinx| + 2x
12) 3x⁵+ 7x⁴ - 2x² - x + 6
13) 1+ x + x²/2! + x³/3! + x⁴/4!
14) √x + 2√x² + 3√x³ + 4√x⁴ + 5√x⁵
15) 5x³/⁵√x² - 3x/³√x⁴ + 7x/⁷√x² + 12 ⁴√/³√x.
16) logₐx + log xᵃ + eˡᵒᵍˣ + logeˣ + e¹⁺ˣ
17) 10ˣ. x¹⁰.
18) (x²+7)(x³+10)
19) √x eˣ
20) x secx log(x eˣ)
21) (1+ sinx)/(1- sinx).
22) 2 cosx/(1- sinx)²
23) (cos x - cos 2x)/(1- cos x).
24) (x³ - 2 + 1/x³)/(x - 2 + 1/x)
25) (eˣ + e³ˣ)/(eˣ + e⁻ˣ)
26) e²ˣ(1+ 2x)
27) √(2x) - √(2/x) + (x+4)/(4 - x)
28) {f(x)}ⁿ
29) √(log x)
30) tan⁵x
31) (tan⁻¹x)²
32) ₑ(ax² + bx +c)
33) ₑ{√(x+1) - ₑ √(x -1)}
34) ₇(x²+ 2x)
35) log(sinx)
36) log(ax²+ bx +c)
37) log(secx + tanx)
38) log ₛᵢₙ ₓ
39) logₑ{x + √(x²±a²)}
40) sin€(x)
41) cos(ax + b)
42) sinx°
43) sinx sin 2x sin 3x
44) sin⁻¹(x/a)
45) cot⁻¹(cosec x+ cotx)
46) log(cos x²)
47) 2 tan⁻¹√{(x - a)/(b - x)}
48) cot⁻¹{√(1+ x²)}
49) cos⁻¹{(1- x²)/(1+ x²)}
50) sin⁻¹{2x/(1+ x²)}
51) tan⁻¹{2x/(1- x²)}
52) tan⁻¹{1/√(x² - 1)}
53) tan⁻¹{(3x - x³)/(1- 3x²)} at x= 1
54) tan⁻¹{cosx/(1- sinx)}
55) cos(sin⁻¹x) + tan(cot⁻¹x)
56) sin(cos⁻¹x) + 1/2 sin⁻¹{2x/(1+ x²)}
57) f(logx) where f(x)= logx
58) tan⁻¹{cosx/(1+ sinx)} + sin(eˣ)
59) logₓ(tanx)
60) log √{(1- cosx)/(1+ cosx)} + aˣ.
61) sinx/(1+ cosx)
62) tan⁻¹√{(1- x)/(1+x)}
63) cos{2 tan⁻¹(cosx)}
64) x√(x² + a²)+ a² log{x+ √(x²+ a²)} at x= 0
65) eᵅˣ sin bx.
66) log₁₀ √{(1- cosx)/(1+ cosx)}
67) tan⁻¹√{(1+ cos 2x)/(1- cos 2x)}
68) ₑ√cotx.
69) ₃√(1+ x + x²)
70) log sec(ax + b)³
71) log{2x + 4 + √(4x² + 16x - 22)}
72) tan log sin(ₑx²)
73) xˣ
74) ₓ(1+ x + x²).
75) (tanx)ˢᶦⁿ ˣ at x= π/4.
76) x ᶜᵒˢˣ sin(logₑx)
77) sin⁻¹{2x/(1+ x²)} w.r.t. cos⁻¹{(1- x²)/(1+ x²)}.
78) ₓsin⁻¹x w.r.t. sin⁻¹x.
79) (tan⁻¹x)/(1+ tan⁻¹x) w.r.t. tan⁻¹x.
80) cos⁻¹{(1- x²)/(1+ x²)} w.r.t. tan⁻¹{2x/(1- x²)}.
81) cos⁻¹{(1- x²)/(1+ x²)} w.r.t. sin⁻¹{2x/(1+ x²)}.
82) log[eˣ{(x -1)/(x +1)}³⁾²].
83) {x³√(x² - 12)}/³√(20 - 3x) at x= 4.
84) [x/{1+ √(1- x²)}]ⁿ.
85) cos⁻¹{(8x⁴ - 8x²+1)}
86) xʸ = eˣ⁻ ʸ
87) log(x²+ x+1)/(x² - x +1)
88) tan⁻¹{(cosx - sinx)/(cosx + sinx)}
89) sin⁻¹{2ax √((1- a²x²)}.
90) cos⁻¹{(3 + 5 cosx)/(5+ 3 cosx)}
91) log√{(a cosx - b sinx)/(a cosx + b sinx)}.
92) sin{2 tan⁻¹√{(1 - x)/(1+ x)}.
93) tan⁻¹[√{(a - b)/(a+ b)} tan(x/2)]
94) 3x⁴ - x²y + 2y³= 0
95) x³ + y³ = 3axy.
96) eˣʸ - 4xy = 2.
97) ₓcos⁻¹x
98) xy = cos(xy) at x= π/2 ,y= 0.
99) xʸ. yˣ = 1.
100) (x+ 2)/{(x -1)(x +5)}
101) ₓcos²x
102) x= log(xy)
103) xʸ+ yˣ = 1.
104) ₐx²
105) (sinx)ᶜᵒˢ ˣ+ (cosx) ˢᶦⁿ ˣ.
106) log(xy)= eˣ ⁺ʸ + 2
107) (cosx)ʸ = (siny)ˣ.
108) xᵖ yᑫ = (x + y)ᵖ⁺ᑫ
109) ax² + by² + 2hxy + 2gx + 2fy + c= 0.
110) x= at², y= 2at.
111) x= a cos k,.y = b sin k.
112) x= sin²t, y = tan t.
113) x= a(2t + sin 2t), y = a(1- cos 2t).
114) sinx= 2t/(1+ t²), tan y = 2t/(1- t²).
115) tan⁻¹{t/√(1- t²)} and x= sec⁻¹{1/(2t² - 1)}.
116) y= sin⁻¹(3t - 4t³), x= sec⁻¹{1/(1- 2t²)}
117) y= ₑsin⁻¹t, x= ₑ⁻ cos⁻¹t,
118) y= a sin³t, x= a cos³t.
119) y= a(sint - t cos t), x= a(cost + t sin t) at t= 3π/4.
120) tan y= 2t/(1- t²), cosx = (1- t²)/(1+ t²).
121) x= 3at/(1+ t³), y= 3at²/(1+ t³).
122) [{tanx}ᵗᵃⁿˣ] ᵗᵃⁿ ˣ at x=π/4
Prove:
1) If cos y = x cos(a+ y), then show dy/dx= (cos²(a+ y))/sin a.
2) If sin y = x sin(a+ y), then show dy/dx= (sin²(a+ y))/sin a = sin a/(1- 2x cos a + x²)
3) If √(1- x²) + √(1 - y²) = a(x - y) then show dy/dx= √{(1- y²)/(1- x²)}
4) If f(x)= {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺ ²ˣ then show f'(0) = {2 log(a/b) + (b² - a²)/ab}{a/b}ᵃ⁺ᵇ.
5) If y= x²/2 + x/2 √(x²+ 1) + log✓√{x + √(x²+1)}] then prove 2y = x dy/dx + log(dy/dx)
6) If x= a sin 2t(1+ cos 2t), y= a cos 2t(1- cos 2t), then show 1+ (dy/dx)² = sec²t
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