Calculus Questions

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4 3 2 3 3 O 6 4 The graph of a function y f x is shown Make a sketch of the antiderivative F given that F is continuous and F 0 3 6 ont Det

2 x 5x16 x 4 x 2 x 3 xx2 x 2 Hole vertical A V A X2 y int oro 6 0 0 4 3 Z 1 S 0 1 5 Horizontal Slant Aysymptate x 45x SXt6 X int x 2 x 3 x x 2 0 x 2 x 3 0 X 2 2 X 3 x 2 0 3 0 0 x 3

Next question X Suppose f x 12x Determine and graph four distinct antiderivatives of f 10 8 6 4 2 Clear All Draw 10 8 6 4 2 2 4 6 8 Get a similar question YOU 10 2 4 6 8 16 retry COSES

The figure below shows f x 2 S 4 24 2 3 23 LA te 4 GO O If F x f x and F 0 0 find F b for b 1 3 6 and 8 b 1 3 6 8 F b

Question 2 The graph of h is given below 6 5 4 3 h x dx h x dx h x dx 6 2 h x dx h x 8 dx 1 6 5 4 3 2 1 Use the graph to evaluate each definite integral dx h ar dz h z dx 2 3 6 1 2 3 6

Find the area of the region bounded by the graphs of y e y e 2 x 1 and x 2 using integration Answers should be exact As a suggestion graph the equations you can see the region for which you are solving for the area Area of region A

h z h x dx such that A expresses the area of the region between the functions y e y e72 1 and z 0 Evaluate the integral to find the area of the region Answers should be exact As a suggestion graph the equations so you can see the region for which you are solving for the area Write an integral in the form A a lower limit of integration 0 b upper limit of integration 0 2 h 2 x 7x 1 Area of region A 0 53279

9 The graph of a function g is shown It is known that g has only real zeros a Identify the degree of g b How many turning points does g have c State the zeros of g and their multiplicity d Write the equation for g x in intercept form e Find one other ordered pair on the graph of g that is neither an z nor a y intercept 2 41 104 8 7 6 0 0 2

8 Consider the function h z 3 x 4 x 1 x 1 a Identify the degree of h b How many turning points does h have c Find the y intercept of h d State the zeros of h and their multiplicity e Find one other ordered pair on the graph of h that is neither an z nor a y intercept

4 Write an equation of a polynomial with a intercepts at x 6 x 2 and x 1 and a y intercept of 4

oint Find the area of the region in the xy plane bounded above by the graph of the function f x 1 6 x 4 3 below by the x axis on the left by the line 8 and on the right by the line 26 The area is

The answer above is NOT correct A farmer must plant a rectangular garden comprised of subsections in a grid pattern 10 long and 10 wide Additionally she must bury irrigation pipes around the perimeter and between each of the sections If she has 1254 meters of irrigation pipes what is the maximum of the whole garden

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions ther dy dy du dx du dx Y Y dy Note addition formula to execute next assignment X U 1 x 10 d dx dy du U 1 x du du dy du 28 y ax a In a dx love 10 1 x 2 30 G y e e u e e u y 2X 5

6 Sketch a function for which each of the following conditions is satisfied lim f x 1 f 0 0 x x f x 0 for x 0 f x 0 for 1 x 1 f x 0 for x 0 f x 0 for x 1 and x 1

7 Given the following graph of f x there are local extrema at xX X3 and xs There are points of inflection at x an X4 State the intervals in which the following conditions hold X 4 X 31 O y f x X X 1 a f x 0 f x 0 b f x 0 f x 0 8 If f x x ax b has a turning point at 2 3 determine the values of a and b

Find the area between the curves f x 4x 16x in dark blue and g x 3x 12x in dark red over the interval 1 5 The area between the curves is NOTE The graphs of the functions given intersect on the given interval a

5 Evaluate the definite integral given above Enter your answer as an exact fraction if necessary 51 31 21 8 dt

curves Below we see a region bounded by two curves graph not necessarily to scale The region shaded in light blue is bounded by two curves y 2 x 1 9 in dark red and y 3 x 3 3 in dark blue 150 Part 1 Suppose that we wish to integrate with respect to z to find the value of the shaded area Fill in the blanks so that the resulting integral with respect to will describe the area of the shaded region Part 2 Note Set up the integral so that the lower limit of integration is less than the upper limit of integration Finally after evaluating the integrals above we find that the area of the shaded region equals

Finding the area between curves Below we see a region bounded by two curves graph not necessarily to scale The region shaded in light blue is bounded by two curves y x in dark red and y 2 in dark blue Part 1 Suppose that we wish to integrate with respect to z to find the value of the shaded area Fill in the blanks so that the resulting sum of integrals with respect to 2 will describe the area of the shaded region 68 Note Set up the integral so that the lower limit of integration is less than the upper limit of integration Part 2 Finally after evaluating the integrals above we find that the area of the shaded region equals

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product O F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions then dy dy du dx du dx Note addition formula to execute next assignment 0 dx ax a In a 18 h t 1 t 1 4 20 v x 1 x 2

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions the dy dy du dx du dx Note addition formula to execute next assignment d dx ax ax In a 2 y 4 3x 2 4 y tan sin x cin of

9 Determine if each statement is true or false Explain and justify your answer a If f 2 0 then f x as a local maximum or a local minimum when x 2 b If f x is a polynomial of degree 3 then its graph has one possible point of inflection

2 Which values for x below are solutions to the inequality 5 s 2 a 10 b 20 c 2 2 d 6 e 9 x NX

D D 3 D a State the domain b Determine the intervals of increase and decrease Increasing c Determine all local maximum and minimum points Local maximum d Determine the intervals of concavity Concave up e State any points of inflection Decreasing Local minimum Concave down YA

3x 3 Determine the equations of all asymptotes for the function f x 4x 7x 1 x 2x 1

2 Match each equation with the graph of the corresponding function below A y B y X x 3 4x x 9 C y D y x x 3 x 1 X 1 8 20 HITHTHITHI HH 2 4 6 8 HHH 4 2 11

4 For the function f x x 4 x a State the domain cp pat 491 04 00 00 194

Calculate the following limits x 3 x 2 x 1 a limx 3x 2x 1 c limx 2x 5x 7 e limx 4x 3x 2x 3x 5 g limx sin 2x X 2x 5 x 1 3x 2 b limx d limx f limx 4x 2x 5 2x3 3x 1 5x 3x 2 3x 2x 1 2 x 1 h limx x 2

If possible approximate a root of the following functions up to 1 decimal place in the given intervals For this exercise you are allowed to use a calculator 1 Function f x x 2x 5 g x sin x x 1 h x x53x 1 j x x x 12 k x x x 2x x 1 m x cos x x n x x55x 4x 1 p x tan x x 1 q x x 4x 3x 2 Interval 0 4 0 2 0 1 0 30 1 2 0 1 1 2 0 4 2 1

2 If possible approximate a root of the following functions up to 1 decimal place in the given intervals For this exercise you are allowed to use a calculator 1 Function f x x 2x 5 g x sin x x 1 Interval 0 4 0 2

Determine whether the following functions to R are injective surjective c bijective a f x 2x 1 d k x x 2 g p x x b g x x 3 e m x x h q x x x 1 1 c h x x 1 x 1 i r x 3x 1 f n x

4 Find an inverse for the following functions Be careful Some of this func tions are not injective so determine a maximal domain where the inverse is defined a f x 3x 2 d k x x 3 x 1 b g x e m x c h x x 4 21 f n x x4 5 when x 0 x 3

2 Calculate go f and fog for the following f g a f x 2x 3 g x x 1 d f x g x x 2 g f x 1 x 1 g x x 4 b f x x c f x x 2x 1 1 g x 3x 5 g x f f x x 2 g x x 2 i f x x 1 g x x 3 e f x 2 x 1 g x x 4 h f x x 1 g x x 2x X

5 Consider the function Find all the points where f is continuous f x 1 if x Q 0 if x Q

3 Sketch the following sets of reals a The set A represented by the union of intervals 1 2 and 4 6 b the set B represented by the intersection of intervals 3 1 and 2 5 c the set C represented by the union of intervals 3 2 and 4 d the set D represented by the intersection of intervals 1 and 0 5 e the set E represented by the union of the interval 1 4 and the set of rational numbers in the open interval 4 6

3x 4x 5x

f Determine if the function h x if x 0 0 if x 0 is continuous at x 0 if x 0 8 X

4 Find all the points where x5 x 5x x 8x 4 0 Hint T polynomal has exactly two solutions namely 2 and 1

3 Let f x x 2x 2x8 x 2 What is the domain of f Is f continuous at 4 Can we define f 4 to make f continuous at 4 If so what is th

X 2 b Investigate the continuity of the function g x at x 2 c Check the continuity of the function h x x 3 at x 3

x sinx if x 0 h Investigate the continuity of the function f x 0 if x 0 x 0 at

Find all x ER satisfying the following inequalities X a x 3 2x 5 0 b 3x 4x 5x2 c 2x 1 3 5x7 2x 4 2 x 3 e 2x 1 3 9 x x 3 0 d f X 0

d Investigate the continuity of the function f x x 1 X 1 x if x 0 if 0 x 1 at if x 1

1 Solve the following exercises about continuity Do not forget to check the right limit as well as the left limit when necessary a Determine if the function f x 2x 3x 1 is continuous at x 2

For f x 9x 9 and g x x find the following composite functions and state the domain of each a fog b gof c fof d gog answers as needed B The domain of fog is all real numbers www b gof x 81x 162x 81 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain of g of is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B The domain of g of is all real numbers c fof x 81x 90 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain off of is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain of f of is all real numbers

For f x 9x 9 and g x x find the following composite functions and state the domain of each a fog b gof c fof d gog ww a fog x 9x 9 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain of fog is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B The domain of f o g is all real numbers b gof x Simplify your answer

Given f x 7 x and g x 6x find the following expressions a fog 4 b gof 2 c fof 1 d gog 0 www a fog 4 14 6 Type an exact answer using radicals as needed Simplify your answer b gof 2 42 2 Type an exact answer using radicals as needed Simplify your answer c f of 1 Type an exact answer using radicals as needed Simplify your answer

x 4 x 1

h X 2 x 3 1 2