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deriv review [71 marks]
1a. A function is given by. Write down the derivative of.
f f ( x ) = 4 x^3 + − 3 , x ≠ 0
3 x^2
f
1b. Find the point on the graph of at which the gradient of the tangent is equal to 6.
f
2a. Maria owns a cheese factory. The amount of cheese, in kilograms, Maria sells in one week, , is given by , where is the price of a kilogram of cheese in euros (EUR). Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.
Q
Q = 882 − 45 p
p
2b. Maria earns for each kilogram of cheese sold. Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.
( p − 6.80)EUR
2c. To calculate her weekly profit , in EUR, Maria multiplies the amount of cheese she sells by the amount she earns per kilogram. Write down an expression for in terms of.
W
W p
2d. Find the price, p , that will give Maria the highest weekly profit.
3a. A function is given by. Find the exact value of each of the zeros of.
f f ( x ) = ( 2 x + 2 )( 5 − x
2
f
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3b. Expand the expression for f ( x ).
3c. Find f ′( x ).
3d. Use your answer to part (b)(ii) to find the values of for which is increasing.
x f
3e. Draw the graph of for and. Use a scale of 2 cm to represent 1 unit on the -axis and 1 cm to represent 5 units on the -axis.
f − 3 ⩽ x ⩽ 3 − 40 ⩽ y ⩽ 20
x y
3f. The graph of the function intersects the graph of. Write down the coordinates of the point of intersection.
g ( x ) = 5 x^ + 6 x − 6 f
4a.
Consider a function. The line L with equation is a tangent to the
graph of when Write down.
f 1 y = 3 x + 1
f x = 2
f ′^ ( 2 )
4b. Find f ( 2 ).
4c. Let and P be the point on the graph of where.
Show that the graph of g has a gradient of 6 at P.
g ( x ) = f ( x^2 + 1 ) g x = 1
4d. Let L be the tangent to the graph of g at P. L intersects L at the point
Q.
Find the y-coordinate of Q. 2 1 2
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Printed for LINCOLN SCH COSTA RICA © International Baccalaureate Organization 2022 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional® 7a. Consider the function.
Find f'( x)
f ( x ) =
x^4 4
7b. Find the gradient of the graph of f at x = − 1.
2
7c. Find the x-coordinate of the point at which the normal to the graph of f
has gradient −.
1 8
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