CALCULUS 1000 B – WINTER 2022
Total: 45 marks NOTE: SHOW ALL YOUR WORK FOR THE PROBLEM ON THIS PAGE.
UNJUSTIFIED ANSWERS MAY RECEIVE LITTLE OR NO CREDIT.
Questions (1) (6 marks) The graph of f(x ) is given. On the right template, sketch the graph of the corresponding derivative f0 (x ):
(a) (b) 1 2 (2) (4 marks) If it is known that f(7) = 0 andf0 (7) = 10 , …ndlim x ! 0f (7 + 3 x) + f(7 + 5 x) x :
Hint: use the de…nition of a derivative.
(3) (4 marks) Find values of A,B and Csuch that the given function is di¤erentiable every- where.
h(x ) = 8 > < > : sin( A(x 1)) x +1 x > 1 B x = 1 C p x + 1 x <1 (4) (4 marks) Ifk 1;the graphs of y= sin xand y= ke x intersect at some points for x 0:
(a)Find the smallest value of kfor which the graphs of these two curves are tangent to each other.
(b)Find x and y coordinates of the point of tangency.
(5) (6 marks) The variable yis a function whose dependence on xis given by the equation cos( x+ 2 y) = 2 x 4y .
(a)Find y0 using the implicit di¤erentiation.
(b)Find an equation of the tangent line at point 4 ; 8 :
(6) (6 marks) Di¤erentiate the given functions (no simpli…cation is required).
(a) f(x ) = ( x + 1) 4 sin( x) log 2( x )(3 x2 1)5 (Hint: use the logarithmic di¤erentiation.) (b) y= (sin x)1 = ln x (7)( 4 marks ) The volume of a right circular cylinder of radius rand height his V = r2 h .
At a certain instant of time, the radius and height of the cylinder are 5 cm and 20 cm, and the volume and the height are increasing at the rate of 500 cm 3 /sec and 4 cm/sec, respectively. How fast is the radius of the cylinder increasing?
(8) Use the following function f(x ) = xe 1=x for answering questions (a) – (f ).
(a) (2 marks) Find intervals where the function f(x ) is increasing and where it is decreas- ing.
(b) (1 mark) Find all critical numbers of f(x ).
(c) (2 marks) Find all local maxima and minima of f(x ) and calculate the values of f(x ) at these points.
(d) (1 mark) Find all in‡ection points of f(x ).
(e) (2 marks) Find the intervals where f(x ) is concave upward and where it is concave downward.
(f ) (3 marks) Sketch the graph of the function f(x ). 3
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