BTEC NATIONAL CERTIFICATE MATHEMATICS for ENGINEERING UNIT 4 ASSIGNMENT 5 - DIFFERENTIATION

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1 BTEC NATIONAL CERTIFICATE MATHEMATICS for ENGINEERING UNIT 4 ASSIGNMENT 5 - DIFFERENTIATION NAME: Time allowed 3 weeks from date of issue. I agree to the assessment as contained in this test. I confirm that the work submitted is my own work. Date issued Date submitted: Signature: Grading criteria PASS Ass1 Ass Ass3 Ass4 Ass5 Ass 6 manipulate and simplify algebraic, logarithmic and exponential functions use standard formulae to find surface areas and volumes of regular solids solve triangular and circular measurement problems involving use of the sine, cosine, tangent and radian functions use graphical methods to produce answers to simple problems involving algebraic, trigonometric and oscillatory functions manipulate statistical and scientific data, and produce statistical diagrams and graphical solutions from such data produce answers to statistical problems involving the determination of mean, median and mode differentiate polynomial and other simple algebraic expressions use the rules of integration to find indefinite and definite integrals of basic polynomial functions. MERIT apply algebraic laws and trigonometric functions to the solution of realistic engineering problems Ass1 Ass Ass3 Ass4 Ass5 Ass 6 apply statistical methods to the analysis of statistical, scientific and experimental data and make realistic estimates and predictions from such an analysis produce answers to a problem involving the determination of the standard deviation and variance use graphical methods to find the differential coefficient of simple exponential and sinusoidal functions differentiate algebraic, exponential and trigonometric functions using the basic rules. DISTINCTION solve realistic engineering problems which involve the mathematical manipulation and analysis of relatively complex algebraic, exponential and trigonometric functions Ass1 Ass Ass3 Ass4 Ass5 Ass 6 apply graphical methods to the solution of engineering problems that involve exponential growth and decay, logarithmic and sinusoidal functions apply the rules for definite integration to engineering problems that involve summation.

2 Feedback Comments: Grade Awarded: Tutor Signature: Date:

3 STANDARD DIFFERENTIALS LOOK UP TABLE y = sin(ax) y = cos(ax) y = tan(ax) y = ln(ax) y = ae kx = acos(ax) = asin(ax) = a + atan(ax) 1 1 = x = x kx = ake Students doing this as an assignment should not use the same material unless it is done as a time constrained test with all students doing it at the same time. Some extra material is included to allow variation by the tutor. INSTRUCTIONS There are four parts to this assignment. You must do as much as you can in the time allocated. Grading depends on the accuracy of your work, the quality of your presentation and the depth of comprehension that you demonstrate. You are allowed THREE WEEKS to complete the work.

4 PART 1 1. Write down the first and second differential coefficients of the following functions. The example shows what is required. EXAMPLE SOLUTION u = v 5 + 3v du 4 = 10v + 3 dv d dv u = 40v 3 i. p = q 3 + 3q 4q +3 ii. y = 0x 6 3x 4 + 5x iii. y = sin(3t) iv. x = 4 cos (5t) v. u = 3 ln v vi. x = 3 e θ PART When a projectile is fired vertically, the height is related to time by the following equation. h = 00t 5t The velocity at any time is v = dh/dt h is measured in metres and t in seconds. (i) Write the equation for the velocity of the projectile. (ii) What is the velocity when t = 0? (iii) What is the velocity when t = 4? (iv) What value of t makes the velocity zero? PART 3 t 3. The electric current flowing in an inductor is related to time by i = ( 1 e ) (i) Plot 'i' vertically against time 't' horizontally over the range t = 0 to t = 4. (ii) Determine the gradient of the graph at t = 1 by finding the gradient of the chord between the points t = 1 and t = 1. (iii) Differentiate the current and solve di/dt at t = 1 and compare the answer to the answer in part (ii) PART 4 A mechanism connected to the shaft of a motor moves with a reciprocating motion such that the distance 'x' moved from the mean position at any time 't' is given by x = 50 0 sin(ωt) where ω is the angular velocity of the shaft. Distance is measured in mm and time in seconds. Given ω = rad/s determine: (i) The speed of the engine in rev/s (ii) The maximum and minimum values of x. (iii) The velocity of the mechanism at any time is v = /dt. (iv) Write down the equation for the velocity of the mechanism. (v) Plot velocity against time over the range 0 to 4 seconds and determine from the graph the maximum velocity of the mechanism and the time at which it occurs the first time. (vi) What is the displacement at the time found in (v)?

5 VARIATIONS ON PART 1 i p = 3q 3 + q +3 ii y = 10x 5 3x 3 + 5x - iii y = 3sin(5t) iv x = 5 cos (t) v u = 6 ln v vi x = 4 e 5θ i. p = q 3 + 3q +3q ii. y = x 7 x + x iii. y = 6sin(3t) iv. x = 3 cos (3t) v. u = 4 ln v vi. x = e θ i. p = 5q 4 + 3q 3 +3q ii. y = 7x 5 5x 3 + x iii. y = 5sin(5t) iv. x = cos (4t) v. u = 1 ln v vi. x = 4 e θ i. p = q 4 + q 3 +6q ii. y = 3x 5 x 3 + x iii. y = sin(t) iv. x = cos (5t) v. u = 5 ln v vi x = 5 e 3θ VARIATION FOR PART The deflection of a simply supported beam y (µm) is related to the distance from the end x (metres) by the following formula. y = 00x 3 100x The. (i) Write down the formula for the gradient of the beam is / at any point (ii) What is the gradient when x = 0? (iii) What is the gradient when x=? (iv) What value of x makes the gradient zero?