MA103 Fall 2013 Midterm - MA103 Midterm Test[10 marks 2013 Fall Page 1 of 2 1 Without using LHpitals Rule evaluate the following limits if they exist o

# MA103 Fall 2013 Midterm - MA103 Midterm Test[10 marks 2013...

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MA103 – Midterm Test 2013 Fall Page 1 of 2 1. Without using L’Hˆopital’s Rule, evaluate the following limits if they exist:[10marks](a) limx2x2x2x2(b)limx5-2x10|x5|(c) limx0x+255x(d)limx→∞4x4+3x23x3/2+22.f(x) is defined asf(x) =braceleftbiggeAxwhenx<0,2x+Bwhenx0.For what values ofAandB[5 marks]willf(x) be continuous and differentiable everywhere?3.(a) State the limit definition of the derivative. [ 3 marks ] (b) Using the limit definition of the derivative, compute the derivative for the function[5 marks]f(x) =x2+ 2x-1.4. For each of the following, findy=dydxusing any method you wish. Leave your answer[12marks]in terms ofxorxandy. Do NOT spend time simplifying your answers.(a)y=x23x(b)y=2tan(x)1+x2(c)y= (sin(x))1/x(d)y2=ex+y+ cos(x2)5. Using L’Hˆopital’s Rule, if it is needed, evaluate the following limits.[5 marks](a) limx0log2(x+1)x(b)limx→∞x2sin(1/x)6. A pile of sand forms a cone (V=1πr2h). Over time, rainfall wears away at the cone,[7 marks]causing the height of the cone to decrease, while maintaining a conical shape. None of