1.07a Derivative as gradient: of tangent to curve

3 The equation of a curve is \(y = ( x - 3 ) \sqrt { x + 1 } + 3\). The following points lie on the curve. Non-exact values are rounded to 4 decimal places. $$A ( 2 , k ) \quad B ( 2.9,2.8025 ) \quad C ( 2.99,2.9800 ) \quad D ( 2.999,2.9980 ) \quad E ( 3,3 )$$

1. Find \(k\), giving your answer correct to 4 decimal places.
2. Find the gradient of \(A E\), giving your answer correct to 4 decimal places.
   The gradients of \(B E , C E\) and \(D E\), rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.
3. State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point \(E\).

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 3 x + 2 ) ^ { 2 } }\). The curve passes through the point \(\left( 2,5 \frac { 2 } { 3 } \right)\).
Find the equation of the curve.

1 The following points $$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$ lie on the curve \(y = \mathrm { f } ( x )\). The table below shows the gradients of the chords \(A E\) and \(B E\).

1. Complete the table to show the gradients of \(C E\) and \(D E\).
2. State what the values in the table indicate about the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

8 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
2. Find the possible \(x\)-coordinates of \(A\).

1 Find the value of the constant \(c\) for which the line \(y = 2 x + c\) is a tangent to the curve \(y ^ { 2 } = 4 x\).

3 A curve has equation \(y = 2 x ^ { 2 } - 6 x + 5\).

1. Find the set of values of \(x\) for which \(y > 13\).
2. Find the value of the constant \(k\) for which the line \(y = 2 x + k\) is a tangent to the curve.

6 \includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x - 3 }\) and \(y = 2 \ln x\). When \(x = a\) the tangents to the curves are parallel.

1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
2. Verify by calculation that this equation has a root between 1 and 2 .
3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-08_351_1031_264_516} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { - \mathrm { ax } }\), where \(a\) is a positive constant, and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 2 } { a } } x e ^ { - a x } d x\).

6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$

1. Find the velocity of the particle at the instant when its acceleration is zero.
   The particle comes to instantaneous rest at \(Q\).
2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
   1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
   3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) does not reach the pulley.]
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(b = 9\) and find the value of \(c\).
2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle \(P\) starts at rest and moves in a straight line from a point \(O\). At time \(t\) s after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = b t + c t ^ { \frac { 3 } { 2 } }\), where \(b\) and \(c\) are constants. \(P\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\) and has velocity \(13.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 9\).

1. Show that \(b = 3\) and \(c = - 0.5\).
2. Find the acceleration of \(P\) when \(t = 1\).
3. Find the positive value of \(t\) when \(P\) is at instantaneous rest and find the distance of \(P\) from \(O\) at this instant.
4. Find the speed of \(P\) at the instant it returns to \(O\).

6 A particle starts from rest at a point \(O\) and moves in a horizontal straight line. The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\). For \(0 \leqslant t < 60\), the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when \(t = 60\), and reverses the direction of its motion. The particle subsequently comes to rest at the point \(A\) when \(t = 100\), and for \(60 < t \leqslant 100\) the velocity is given by $$v = 0.025 t - 2.5$$

1. Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
2. Find the total distance travelled by the particle.
3. Find the maximum speed of the particle and sketch the particle's velocity-time graph for \(0 \leqslant t \leqslant 100\), showing the value of \(t\) for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = x ^ { 2 } + 3 x - 2\) The point \(P ( 3,16 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(3 + h\) also lies on the curve.
2. Find, in terms of \(h\), the gradient of the line \(P Q\). Write your answer in simplest form.
3. Explain briefly the relationship between the answer to (b) and the answer to (a).

6. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { ( x + 3 ) ^ { 2 } } { x \sqrt { x } }\)
• the point \(P ( 4,20 )\) lies on \(C\)
• Find \(\mathrm { f } ( x )\), simplifying your answer.
  

1. A curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\)

Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = 4 x + \frac { 1 } { \sqrt { x } }\)
• the point \(P\) has \(x\) coordinate 4 and lies on \(C\)
• the tangent to \(C\) at \(P\) has equation \(y = 3 x + 4\)
  1. find an equation of the normal to \(C\) at \(P\)
  2. find \(\mathrm { f } ( x )\), writing your answer in simplest form.

7. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation \(y = 2 x ^ { 2 } + 5\) The point \(P ( 2,13 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
2. Find, in terms of \(h\), the gradient of the line \(P Q\). Give your answer in simplest form.
3. Explain briefly the relationship between the answer to (b) and the answer to (a).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)

Given that

• \(f ^ { \prime } ( x ) = \frac { 4 x ^ { 2 } + 10 - 7 x ^ { \frac { 1 } { 2 } } } { 4 x ^ { \frac { 1 } { 2 } } }\)
• the point \(P ( 4 , - 1 )\) lies on \(C\)
  1. (i) find the value of the gradient of \(C\) at \(P\) (ii) Hence find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
  2. Find \(\mathrm { f } ( x )\).

1. The curve \(C\) has equation

$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$

1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
3. Find the \(x\)-coordinates of these two points.

6. (i) The curve \(C _ { 1 }\) has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that \(C _ { 1 }\) has a stationary point at \(x = \frac { \sqrt { p } } { 2 }\) where \(p\) is a constant to be found.
(ii) A different curve \(C _ { 2 }\) has equation $$y = 4 x - 12 \sin ^ { 2 } x$$

1. Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where \(A\) and \(B\) are constants to be found.
2. Hence, state the maximum gradient of this curve.

8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .

1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
2. Prove that P is the mid-point of RS.

8. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 2, has equation $$y = 2 x ^ { \frac { 1 } { 2 } } , \quad 1 \leqslant x \leqslant 8$$

1. Show that the length \(s\) of curve \(C\) is given by the equation $$s = \int _ { 1 } ^ { 8 } \sqrt { } \left( 1 + \frac { 1 } { x } \right) \mathrm { d } x$$
2. Using the substitution \(x = \sinh ^ { 2 } u\), or otherwise, find an exact value for \(s\). Give your answer in the form \(a \sqrt { } 2 + \ln ( b + c \sqrt { } 2 )\) where \(a , b\) and \(c\) are integers.

9

1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
4. Deduce the value of \(k\).

6

1. Find \(\int x \left( x ^ { 2 } + 2 \right) \mathrm { d } x\).
2. 1. Find \(\int \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
   2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } x }\). Find the equation of the curve, given that it passes through the point \(( 4,0 )\).

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve.