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Mathematics Grade 12 Unit 2: INTRODUCTIONS TO CALCULUS

73 national exam questions have come from this unit. It runs 100 pages; one careful study pass takes about 11.0 hours, and its difficulty is moderate among Mathematics units. Each section below lists the questions set from it, with the year, and opens the page that teaches it.

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2.1 Introduction to Derivatives · read p. 63

2018 E.C., Q11 – Let a function, $f(x)=\begin{cases}2x^2-4, & \text{if } x\geq 1\ x^3-4x+1, & \text{if } x<1\end{cases}$. Then what is the equation of the tangent line to the graph of $f$, at $x=-2$?
2018 E.C., Q12 – Let a function, $f(x) = \begin{cases} 2x^2 - 4, & \text{if } x \ge 1 \\ x^3 - 4x + 1, & \text{if } x < 1 \end{cases}$. Then what is the equation of the tangent line to the graph of $f$, at $x = -2$?
2018 E.C., Q46 – Given the exponential functions and , which statement describes their monotonicity?
2018 E.C., Q6 – What is the average rate of change of $y=f(x)$ with respect to $x$ on the interval $[a,b]$?
2018 E.C., Q24 – Let f(x)=2x+1/x–1. Which of the following equations is the equation of the tangent line to the graph of f at (2,f(2))?
2017 E.C., Q51 – The solution set of $\sqrt[3]{64^{x-1}} = \left(\frac{1}{16}\right)^{2x+3}$ is equal to:
2017 E.C., Q7 – What is the domain of the function $f(x) = \sqrt{3 - x^2}$?
2017 E.C., Q55 – The slope of a line passing through and is equal to:
2017 E.C., Q20 – What is the average rate of change of $g(x)=x^3-2x$ over the interval $1\le x\le 6$?
2017 E.C., Q29 – Suppose and be given two functions. Then which of the following is equal to ? 3x + − 2 1 4x + 3 2 x + 3 2 − 3 1 4x + 3 2x
2017 E.C., Q17 – Let $f$ be a function and let $c$ be an element of domain of $f$ at which the gradient of $f$ is zero. Which one of the following statements is true about $f'(c)$?
2017 E.C., Q59 – What is the equation of the tangent line to the graph of $f(x) = 2x^3 - 3x^2 + 1$ at $x = 2$?
2017 E.C., Q2 – Let $f(x)=(1-2x)^{\frac{1}{2}}$. Then what is the values of $f'(x)$?
2016 E.C., Q31 – Which of the following gives the definition of integration for a function?
2016 E.C., Q24 – Which one of the following is the solution set of the equation $(\sqrt{16})^{2x-1} = \left(^3\sqrt{4096}\right)^{3x-5}$?
2016 E.C., Q29 – An enterprise found out its profit by the formula $f(t)=t^2-t+100$, t in days. Which of the following is the average rate of change of its profit on $0\le t\le 2$?
2016 E.C., Q30 – For the purpose of road management, flow of vehicles needs to be regulated. On the average it was observed that the speed of vehicles in km/hr follow the formula $f(x)=10\left(x+\frac{50}{2x+1}\right)$ where $x$ is the s
2016 E.C., Q50 – Let $f$ be a differentiable function at a number 'a' for $a \in \mathrm{Domain}\ f$. In the expression $\frac{f(a+h)-f(a)}{h}$ for $h \ne 0$, as $h$ gets closer and closer to zero from both directions, the expression bec
2015 E.C., Q35 – Which one of the following is true about the derivative of the given functions? d ( x4 x−ln x) , x = x6 −4x +x−1+4x ln x 2 0 (x sin x) = dx d cos x −x sin x
2015 E.C., Q51 – Which of the following describes the mechanical meaning of derivative?
2015 E.C., Q47 – Let $f(x)=\frac{1}{\sqrt{x}}$ and $g(x)=2-\frac{1}{6}x^3$ be two functions. For what values of x do the graphs of f and g have tangent lines of the same slope?
2015 E.C., Q36 – Which one of the following is an equation of the line tangent to the graph of the function $f(x)=x^3-3x^2+x+1$ at $(1, 0)$?
2015 E.C., Q12 – What is the value of x in the equation $\sqrt{125x^{2} 5^{x-1}} = 25^{3x + 1}$?
2015 E.C., Q31 – Which one of the following is NOT true about the derivative of a function?
2015 E.C., Q50 – Let $f$ and $g$ be differentiable functions, and $h(x)=g(f(3x+1))$. If $f'(1)=f(1)=2$ and $g'(2)=1$, then what is the value of $h'(0)$?
2014 E.C., Q34 – Suppose f is a continuous function on an interval [a, b] and differentiable on (a, b) with the property f(a) = f(b). Which one of the following must be true?
2014 E.C., Q33 – Let $f(x)=x^2+\sqrt{x}$ and $g(x)=e^{2x}$. Which of the following is true?
2014 E.C., Q19 – What is the solution set of the equation $3^{(x-x^2)}=\frac{1}{9}$?
2014 E.C., Q34 – What is the equation of the tangent line to the graph of the function $f(x)=e^x$ at $(1,e)$?
2014 E.C., Q32 – Let $f(x)=2\sqrt{x+1}$. Which one of the following indicates the interval and its derivative of f respectively?
2014 E.C., Q31 – Let $f$ be a continuous function on R and $c$ be a number in the domain of $f$. Which one of the following must be true about the function?
2014 E.C., Q35 – Suppose f is a twice continuously differentiable function and f(x) = f(2x + 3). Then the second derivative of g at 0 is equal to:
2013 E.C., Q33 – Which of the following is the same as the derivative of a function f' at a point P, where P is on the graph of f?
2013 E.C., Q37 – Which of the following is true about the function defined by $f(x) = x^3 - 3x^2$ on [-1, 3]?
2013 E.C., Q34 – What is the derivative of the function $f(x) = 3x^2 + 2\sqrt{x} - 4x$ at $x = 1$?

2.2 Application of Derivative · read p. 112

2018 E.C., Q7 – A rectangular metal sheet of dimension 12 m by 18 m is cut into six congruent rectangular sheets (or pieces). Then what is the area of each piece in m²?
2017 E.C., Q33 – A circle with radius 3 cm increases its radius at a rate of 0.2 cm/sec. What is the area of the circle at the 5 second? 9.64π 2 4π 2
2013 E.C., Q54 – The acceleration of a body at any time t is given by $a(t)=4-2t+3t^2$ with initial velocity v(0) = 5. Then what is the velocity v of the body as a function of time?
2013 E.C., Q39 – If the radius $r$ of a sphere is increasing at the rate of 2 cm/min, then the rate of change of the volume when $r = 1 \text{ cm}$ is
2013 E.C., Q40 – If the radius r of a sphere is increasing at the rate of 2 cm/min, then the rate of change of the volume when r = 1 cm is

2.3 Introduction to Integration · read p. 123

2018 E.C., Q19 – Which one of the following is true about definite integrals?
2018 E.C., Q50 – According to the properties of integration, which of the following is equivalent to $\int (3f(x)-g(x))\,dx$?
2016 E.C., Q46 – What is the area of the region bounded by the graph of $f(x)=x^3$ and the x-axis on the interval $[-4, 2]$?
2015 E.C., Q41 – What is the area (in square units) of the regions bounded by the graph of the equation ?
2015 E.C., Q42 – What is the area (in square units) of the region enclosed by the curve , the x-axis, the vertical lines and ?
2015 E.C., Q27 – If , , and are functions defined by , , and respectively, which one of the following is true?
2015 E.C., Q37 – What is the area of the region bounded by the graph of and the x-axis in square units?
2015 E.C., Q46 – Let F be the anti-derivative of f. Then which of the following is NOT true about F and f?
2015 E.C., Q37 – Which one of the following is true about integration and differentiation?
2015 E.C., Q30 – Which one of the following is the correct definition of continuity of a function on the given interval?
2014 E.C., Q20 – Let f be a continuous function on [a, b]. Then which of the following is NOT true?
2014 E.C., Q26 – Let f and g be two functions. Which one of the following statements is true about f or g?
2014 E.C., Q49 – What is the area of the region bounded by the line $y=x$ and the parabola $y=x^2-2x$ on the interval $[0,3]$?
2014 E.C., Q19 – The integral x 1 −3x + 3 e−3x − 3 e−3x
2014 E.C., Q25 – What is the value of $$\int_a^b x e^x dx$$
2014 E.C., Q22 – The integral $ \int \left( \frac{1}{\sqrt{x}} - e^{-3x} + \frac{5}{x^2} \right) \, dx \text{ is equal to:} $
2014 E.C., Q37 – Which one of the following is equal to $\int_1^3 2x\sqrt{x^2+1}\,dx$?
2014 E.C., Q31 – Let f be a function that is continuous at x o . Which of the following is true about f?
2013 E.C., Q44 – What is the area of the region bounded by y = 0 and the graph of $f(x)=4-x^2$ on the interval $[-1, 1]$?
2013 E.C., Q38 – What are the length $L$ and width $W$ of a rectangle with perimeter 10,000m that maximizes the area?
2013 E.C., Q28 – A factory that produces electric cable had sales Birr 1,000,000 the first day. During each successive days the sales increased by Birr 10,000 every day. What is the total sales of the factory in the first 10 days?
2013 E.C., Q39 – What are the length L and width W of a rectangle with perimeter 10,000m that maximizes the area?
2013 E.C., Q62 – Let $f$ be a function defined on a closed bounded interval $[a, b]$. Which of the following describes that $f$ is continuous on $[a, b]$?
2013 E.C., Q43 – What is the area of the region bounded by $y = 0$ and the graph of $f(x) = 4 - x^2$ on the interval $[-1, 1]$?
2013 E.C., Q3 – If $f$ is a continuous function on $[a, b]$ and $F$ is its anti-derivative, then which of the following is true?
2013 E.C., Q53 – Which of the following is true where c is a constant?
2013 E.C., Q43 – Which of the following is equal to $\int(\cos x-2x)dx$ where c is a constant?
2013 E.C., Q41 – Suppose $\int_{1}^{3} f(x) \, dx = 6$ and $\int_{1}^{2} f(x) \, dx = 10$, what is the value of $\int_{2}^{3} f(x) \, dx$ ?
2013 E.C., Q53 – Which of the following is true where $c$ is a constant?
2013 E.C., Q28 – A factory that produces electric cable had sales Birr 1,000,000 the first day. During each successive day the sales increased by Birr 10,000 every day. What is the total sales of the factory in the first 10 days?