Geometric meaning of derivation

1.What is the geometric meaning of the derivative?

Solution:

Using the derivative of the function y = f(x) we can write the equation of the tangent or the equation of the normal to the graph of the function at the point T [xT , yT]

Equation of the Tangent Line:

y−yT=kt(x−xT),kt=f′(xT)y - y_T = k_t (x - x_T), \quad k_t = f'(x_T)

Equation of the Normal Line:

y−yT=kn(x−xT),kn=−1f′(xT)y - y_T = k_n (x - x_T), \quad k_n = -\frac{1}{f'(x_T)}

Point T [x_T, f(x_T)] is the common point of the tangent (or normal) with the graph of the function.

Subtangent and Subnormal:

The tangent, the normal, and the x-axis form a right triangle Δ ABT (right angle at vertex T).

It holds:

  • Point T lies on line t,

  • Point A lies on line t ∩ x-axis,

  • Point B lies on line n ∩ x-axis.

  • Point T₁ [x_T; 0] is the orthogonal projection of point T onto the x-axis.

Subtangent Sₜ is the length of segment ∣AT1∣|AT_1|:

St=∣yy′∣S_t = \left| \frac{y}{y'} \right|

Subnormal Sₙ is the length of segment ∣T1B∣|T_1B|:

Sn=∣y⋅y′∣S_n = |y \cdot y'|

Length of the Tangent Segment TA = t:

t2=yr2+St2t^2 = y^{2} + S_t^{2

Length of the Normal Segment TB = n:

n2=yr2+Sn2n^2 = y^{2} + S_n^{2}

2. Write the equation of the tangent and the equation of the normal to the curve:

geometricky-vyznam-derivacie-2z

Solution:

geometricky-vyznam-derivacie-2r

equation of tangent                     equation of normal

3. Write the equation of the tangent and the equation of the normal to the curve:

geometricky-vyznam-derivacie-3z

Solution:

geometricky-vyznam-derivacie-3r

equation of tangent                      equation of normal

4. Write the equation of the tangent and the equation of the normal to the curve:


geometricky-vyznam-derivacie-4z
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5. Write the equation of the tangent and the equation of the normal to the curve:

geometricky-vyznam-derivacie-5z
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6. Write the equation of the tangent and the equation of the normal to the curve y = ln(x+1) at the point T[0; yT]

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7. Write the equation of the tangent to the curve y = sin 2x at the point T [3π/4 ; yT].

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8. Write the equation of the tangent to the curve y = x2 – 4x + 3 , which makes an angle φ = 450 with the x-axis.

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9. Find the equation of the tangent to the curve y = x2 - 2x + 3 , if the tangent is parallel to the line p : 3x - y + 5 = 0.

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10.The function is given f : y = x3 - 9x2 + 15x + 3. Determine the contact points of the horizontal tangents.

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11. Determine the angle φ between two tangents of the curve if one has contact point T1[3; yT1] and the other T2[-3; yT2]. The equation of the curve:

geometricky-vyznam-derivacie-11z
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12. Determine the lengths of the subtangent, subnormal, tangent and normal to the graph of the function at the point T[1; yT]. The function has the equation:

geometricky-vyznam-derivacie-12z
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13. Calculate the lengths of the subtangent, subnormal, tangent and normal to the graph of the function y = 2x at the point T [1; yT].

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