Calculus Questions

YA 9 8 7 6 5 1 1 y f x MA Nigh y g x 2 3 4 5 6 7 89 EXAMPLE 4 Using graphs to evaluate composite functions Use the graphs of fa g in Figure 1 9 to find the following values a f g 3 b g f 3 c f f 4 d f g f 8 e g g 9 f g f g 6 X

Find the equation of a line tangent to the curve xy xy x 1 at the point 1 2 OA y 5 667 2 667x OB y 2 x OC y 2 667 1 333x OD y 2 667 1 667x O E y 5 3x

7 10 points The following figure shows the level curves of z two variables P where fr is positive Q where fr is negative 0 R where fy is zero S where fy is negative 10 20 f x y for f x y a function of draw and label points with the following properties If it is not possible to draw such a point don t do so 30 40

21 32 Find the derivative of the function using the definition of derivative State the domain of the function and the domain of its derivative 21 f x 3x 8 22 f x mx b

3 A rechargeable battery is plugged into a charger The graph shows C 1 the percentage of full capacity that the battery reaches as a function of time elapsed in hours a What is the meaning of the derivative C 1 b Sketch the graph of C 1 What does the graph tell you Percentage of Jull charge 100 80 60 40 20 t hours

2 10 pts Define your variables translate the following information into a system of linear equations and then solve the system Write your answer as a complete sentence Two rainstorms occurred in one week in a certain area In the first storm 15 mL of rain fell per hour and in the second storm 30 mL of rain fell per hour The rain lasted for a total of 70 hours resulting in a total rainfall of 1500 mL What was the duration of each storm

1 2 Use the given graph to estimate the value of each derivative Then sketch the graph of f 1 a f 0 e f 4 b f 1 f f 5 ya 1 0 1 y f x c f 2 g 6 X d f 3 h f 7

f x is continuous for 0 5 x 0 2 and also has the following values X f x 0 5 0 4 0 3 0 2 1 1 1 1 3 1 6 Which of the following may be true about f x I f x may be monotonic for 0 5 x 0 2 II f x may be increasing for 0 5 0 2 III f x may be concave up on 0 5 0 2 A I only B II only C III only D I and III only E I II and III

3 Match the graph of each function in a d with the graph of its derivative in I IV Give reasons for your choices b YA VA I III YA 4 4 4 d II IV 3 YA 0 x

The acceleration m of an object along a straight line is given in the following graph sec2 Acceleration 10 5 1 0 1 Time OA 53 5 meters OB 28 5 meters OC 27 0 meters OD 19 5 meters OE 33 0 meters 2 Assuming that the object has an initial velocity of 5m how far does the particle move during the time 0 t 3 sec E

Given that g is the inverse function of f and f 3 4 and f 3 5 then g 4 O A B O C 3 5 O E 3 5 D 1 3

17 For the function g whose graph is given arrange the follow ing numbers in increasing order and explain your reasoning 0 g 2 g 0 g 2 g 4 0 y g x 2 3 4 www X

If q is positive and increasing for what value of q is the rate of increase of q twelve times that of q A 3 OB 4 5 OC 2 OD 12 OE 36

Using a left endpoint Riemann sum approximation with four equal subintervals the approximate value of fo g x dx is A 8 B 8 C 16 OD 16 01 2 3 4 5 6 7 8 g x 1 2 5 3 1 5 1 25 0 1 5 4 5 X E 13 5

5 10 points A function f x y is differentiable and satisfies f 2 1 3 fr 2 1 0 5 f 2 1 5 Find the equation of the tangent plane to the surface z f x y at the point P 2 1 3 and use it to find the best linear approximation to f 2 1 0 98

3 The displacement in meters of a particle moving in a straight line is given by the equation of motion s 1 1 where t is measured in seconds Find the velocity of the particle at times t a t 1 t 2 and t 3

D a Find the slope of the tangent to the curve y 2 x at the point where x a b Find equations of the tangent lines at the points 1 2 and 9 6 c Graph the curve and both tangents on a common screen

4 12 points Let f r be a scalar valued function of a single variable r satisfying f 0 df Define a function dr 2 f 0 1 where f x g x y cos x y f x Find g 0 3 0 and 9 0 3 0 3

If the function f has a continuous derivative on 0 c then ff x dx A f c f 0 B f c f 0 C f c D I f c f 0 I E f x C

Let f x x 9 X 6 g x X X 6 Express f x g x as a rational function f x g x for x Simplify your answers Use a comma to separate answers as needed

Find the first and second derivatives d dx y 29 x 29 x d 2 dx 29 x

Describe the domain of the function g x 1 6 x The domain of the function consists of all real numbers x where Type an inequality

X 9 x Let g x h g x 1 h x x3 3x 8 Calculate the function h g x

Find an equation of the line such that the points An equation of the line containing the points Type an equation 5 and 3 4 4 are on the line 3 5 and 4 is 5

Differentiate y X x 2 2 x dx 2 11

Let f x 7 X a Find the derivative f x b Find f 2 1 a What is the first step in finding the derivative of f x 7 A Substitute x h for x in the original function OB Rewrite the function in the form f x rx OC Rewrite the function in the form f x x O D Rewrite the function in the form f x Rewrite f x 1 7 X in this form XIL X f x Which rule of derivatives should be used to differentiate functions of this form A The rule that if f x mx b then f x m

6 If f x X find f x h f x and simplify f x h f x

If f x x 5x find f 0 f 4 f 5 and f 8 f 0 Simplify your answer f 4 Simplify your answer f 5 Simplify your answer f 8 Simplify your answer

Determine whether the following limit exists Compute the limit if it exists x 3x 4 X 4 lim X 4 Is the function f x x 3x 4 X 4 OYes O No Does the function need to be defined at x 4 in order for its limit to exist at x 4 defined when x 4 O No OYes To evaluate the limit begin by factoring the numerator x 3x 4 x 3x 4 X 4 Factor completely Do not simplify When f x is defined common binomial factors can be cancelled from the numerator and denominator 11 Simplify your answer

Let f x x 8 h x 4 4x Calculate the following function f x h x f x h x

Factor the following polynomial 4x 32x 64 4x 32x 64

a Find the slope of the tangent to the curve y 2 x at the point where x a b Find equations of the tangent lines at the points 1 2 and 9 6 c Graph the curve and both tangents on a common screen

a Find the slope of the tangent line to the parabola y x 3x at the point 1 2 i using Definition 1 ii using Equation 2 b Find an equation of the tangent line in part a c Graph the parabola and the tangent lin As a check on your work zoom in toward the point 1 2 until the parabola and the tangent line are indistinguishable

15 42 Find the limit or show that it does not exist 4x 3 x 5x 1 15 lim 3t t 3 t t 4t 1 17 lim 1118 rr 118 2 3r 19 lim 21 lim 818 3 4 x 2 x 2 xx 3x 7 16 lim 18 lim 1110 3x3 8x 2 xx 4x 5x 2 20 lim 61 1 15 t 9 21 22 lim 8118 u 1 2u 1 u 2

3 For the function whosc graph is given state the following a lim f x b lim f x x18 x118 c lim f x x 1 e The equations of the asymptotes YA 1 1 d lim f x x 3 X

The quadratic formula is used to solve for x in equations taking the form of a quadratic equation ax bx c 0 b b 4ac 2a Solve for x in the following expression using the quadratic formula 2x 35x 5 1 0 Use at least three significant figures in each answer x quadratic formula x I and x

If dx 1 x xy x 0 and y 4 when x 1 when x 3 y OA 2 381 OB 4 711 OC 3 898 OD 3 898 OE 4 711

2 a Can the graph of y f x intersect a vertical asymptote Can it intersect a horizontal asymptote Illustrate by sketching graphs b How many horizontal asymptotes can the graph of y f x have Sketch graphs to illustrate the possibilities

If the graph of y X C asymptote x 2 then b cannot be equal to A 0 ax b B 7 O C 5 OD 12 O E 10 has a horizontal asymptote y 5 and a vertical

1 Explain in your own words the meaning of each of the following a lim f x 5 818 b lim f x 3 8118

The rectangle shown below has vertices of 0 0 0 2 1 and 0 1 y cos x What percent of the rectangle is shaded OA 63 7 B 36 3 OC 31 4 OD 50 OE 43 2

A particle moves along a line so that at time t where 0 t its position t is given by s t 4 sint 10 What is the acceleration of the 2 particle the first time its velocity equals zero A 3 969 OB 0 745 C 2 550 D 1 323 E 5 197

If y 2 f g x then y O A f g x g x B g x g x O c 2f g x g x D 2 g x g x 2 g x g x O E Cannot be determined

If y 2f g x then y A f g x g x B f g x g x c 2f g x g x D 2 g x g x 2 g x g OE Cannot be determined

Which of the following will not give the instantaneous rate of change of f x with respect to x when x 3 f 3 f 3 0 1 0 1 O A OB lim x 3 x 3 OC lim h 0 D lim I 0 f x f 3 OE lim h 0 f 3 h f 3 h f 3 f x 3 x f 3 h f 3 h 2h

What is the minimum value of f x 3xe OA 3e OB 338 e O C 1 D 0 E f x has no minimum value

Given the values of f x shown in the chart below which of the following could be values for f x at x 11 212 and 3 X 1 2 3 f x 3 5 A B C D E X 2 1 2 31 12 f x 2 4 2 X X f x 2 5 X 1 X f x 2 f x 4 3 13 12 2 2 1 2 21 1 2 31 1 2 8 3 2 4 2 12 4 2 2 3 2

Given that f 0 5 2 and f 0 5 4 using a tangent line approximation you would estimate f 0 to be A 0 B 3 OC 2 D 1 E 4 E

A 17 m Using trapezoids estimate the total distance the runner travels from t 0 to t 6 seconds B 12 m C 24 m D 26 m t seconds 0 2 46 2 5 8 12 E 40 m v t m sec

The graph of which of the following equations has y 1 as an asymptote O A y sin x B y In x 3 2x OC y 2x 13x 7 OD y O E y ex D y 1