1.07i Differentiate x^n: for rational n and sums

726 questions

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OCR MEI C2 2013 June Q9
11 marks Standard +0.3
Fig. 9 shows a sketch of the curve \(y = x^3 - 3x^2 - 22x + 24\) and the line \(y = 6x + 24\). \includegraphics{figure_9}
  1. Differentiate \(y = x^3 - 3x^2 - 22x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places. [4]
  2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = -4\). Find algebraically the \(x\)-coordinate of the other point of intersection. [3]
  3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6x + 24\) for \(-4 \leq x \leq 0\), shown shaded on Fig. 9. [4]
OCR MEI C2 2014 June Q11
13 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows a sketch of the curve with equation \(y = x - \frac{4}{x^2}\).
  1. Find \(\frac{dy}{dx}\) and show that \(\frac{d^2y}{dx^2} = -\frac{24}{x^4}\). [3]
  2. Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum. [5]
  3. Find the equation of the normal to the curve when \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [5]
OCR MEI C2 2016 June Q1
5 marks Easy -1.3
  1. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) when \(y = 6\sqrt{x}\). [2]
  2. Find \(\int \frac{12}{x^2} \mathrm{d}x\). [3]
Edexcel C2 Q6
9 marks Moderate -0.3
$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$
  1. Find \(f'(x)\) and \(f''(x)\). [3]
  2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
  3. Determine whether the turning point is a maximum or minimum point. [2]
Edexcel C2 Q3
6 marks Moderate -0.3
Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [6]
Edexcel C2 Q6
9 marks Standard +0.3
A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),
  1. find the values of \(a\) and \(b\), [6]
  2. find the coordinates of the other stationary point of the curve. [3]
Edexcel C2 Q8
13 marks Standard +0.3
The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
  2. Find the exact coordinates of the stationary point of \(C\). [5]
  3. Determine the nature of the stationary point. [2]
  4. Sketch the curve \(C\). [2]
OCR MEI C2 Q1
4 marks Easy -1.2
Differentiate \(x + \sqrt{x^3}\). [4]
OCR MEI C2 Q4
4 marks Moderate -0.3
Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]
OCR MEI C2 Q6
2 marks Easy -1.8
Differentiate \(10x^4 + 12\). [2]
OCR MEI C2 Q7
12 marks Moderate -0.3
\includegraphics{figure_7} Fig. 10 shows a solid cuboid with square base of side \(x\) cm and height \(h\) cm. Its volume is \(120\) cm\(^3\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A\) cm\(^2\), of the cuboid is given by $$A = 2x^2 + \frac{480}{x}.$$ [3]
  2. Find \(\frac{dA}{dx}\) and \(\frac{d^2A}{dx^2}\). [4]
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case. [5]
OCR MEI C2 Q8
2 marks Easy -1.2
Differentiate \(6x^{\frac{5}{2}} + 4\). [2]
OCR MEI C2 Q9
5 marks Moderate -0.8
A is the point \((2, 1)\) on the curve \(y = \frac{4}{x^2}\). B is the point on the same curve with \(x\)-coordinate \(2.1\).
  1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places. [2]
  2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A. [1]
  3. Use calculus to find the gradient of the curve at A. [2]
OCR MEI C2 Q2
5 marks Moderate -0.3
Find the equation of the normal to the curve \(y = 8x^4 + 4\) at the point where \(x = \frac{1}{2}\). [5]
OCR MEI C2 Q3
13 marks Moderate -0.3
  1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
  2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
    1. Expand \((2 + h)^4\). [3]
    2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
    3. Show how your result in part (iii) (B) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]
OCR MEI C2 Q4
12 marks Moderate -0.8
  1. Calculate the gradient of the chord joining the points on the curve \(y = x^2 - 7\) for which \(x = 3\) and \(x = 3.1\). [2]
  2. Given that \(f(x) = x^2 - 7\), find and simplify \(\frac{f(3 + h) - f(3)}{h}\). [3]
  3. Use your result in part (ii) to find the gradient of \(y = x^2 - 7\) at the point where \(x = 3\), showing your reasoning. [2]
  4. Find the equation of the tangent to the curve \(y = x^2 - 7\) at the point where \(x = 3\). [2]
  5. This tangent crosses the \(x\)-axis at the point P. The curve crosses the positive \(x\)-axis at the point Q. Find the distance PQ, giving your answer correct to 3 decimal places. [3]
OCR MEI C2 Q1
13 marks Moderate -0.3
\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).
  1. Write down \(\frac{dy}{dx}\). [2]
  2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
  3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
  4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]
OCR MEI C2 Q2
11 marks Standard +0.3
Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]
OCR MEI C2 Q3
12 marks Moderate -0.3
The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).
  1. Sketch the curve. [2]
  2. Use calculus to find the equation of the tangent to the curve at A. [4]
  3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]
OCR MEI C2 Q4
12 marks Moderate -0.3
\includegraphics{figure_3} A is the point with coordinates (1, 4) on the curve \(y = 4x^2\). B is the point with coordinates (0, 1), as shown in Fig. 10.
  1. The line through A and B intersects the curve again at the point C. Show that the coordinates of C are \(\left(-\frac{1}{4}, \frac{1}{4}\right)\). [4]
  2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = -2x - \frac{1}{4}\). [6]
  3. The two tangents intersect at the point D. Find the \(y\)-coordinate of D. [2]
OCR MEI C2 Q5
5 marks Moderate -0.5
Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]
OCR MEI C2 Q2
11 marks Moderate -0.3
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
OCR MEI C2 Q8
5 marks Easy -1.3
  1. Differentiate \(12\sqrt{x}\). [2]
  2. Integrate \(\frac{6}{x^5}\). [3]
Edexcel C3 Q6
10 marks Standard +0.3
The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$
  1. Find the value of f''(x) at \(x = 4\). [3]
  2. Given that f(3) = 0, find f(x). [4]
  3. Prove that f is an increasing function. [3]
Edexcel C3 Q1
5 marks Moderate -0.8
The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]