Calculus Review Sheet 3

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of Calculus 1.

(1) Identify the inflection points and local maxima and minima of the function Identify the intervals on which the function is concave up and concave down.  Sketch the graph showing these specific features.

(2) Identify the inflection points and local maxima and minima of the function   Identify the intervals on which the function is concave up and concave down.  Sketch the graph showing these specific features.

(3) Sketch the graph of the following function Find all vertical and horizontal asymptotes of the graph of each function. Determine intervals of increasing and decreasing, determine concavity, and locate all critical points and points of inflection. Show all special features such as cusps or vertical tangents.

(4) Find constants and that guarantee that the graph of the function defined by will have a vertical asymptote at and a horizontal asymptote at Sketch the graph of the  function.

(5) Explain why the function has a vertical asymptote but no vertical tangent. Sketch the graph of the function.

(6) The bottom of an 8-ft-high mural painted on a vertical wall is 13 ft above the ground. The lens of a camera fixed to a tripod is 4 ft above the ground. How far from the wall should the camera be placed to photograph the mural with the largest possible angle?

(7) Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a sphere of radius

(8) A truck is 250 mi due east of a sports car and is traveling west at a constant speed of 60 mi/h. Meanwhile, the sports car is going north at 80 mi/h. When will the truck and the car be closest to each other? What is the minimum distance between them?

(9) A jeep is on the desert at a point located 40 km from a point , which lies on a long straight road. The driver can travel at 45 km/h on the desert and 75 km//h on the road. The driver will win a prize if he arrives at the finish line at point , 50 km from , in 84 minutes or less. What route should he travel to minimize the time of travel? Does he win the prize?

(10) Use L'Hospital's rule to find the limit  

(11) Use L'Hospital's rule to find the limit  

(12) Use L'Hospital's rule to find the limit

(13) Use L'Hospital's rule to find the limit   

(14) For a certain value of the limit is finite and nonzero. Find and then use L'Hospital's rule to compute the limit.

(15) Find an antiderivative for the functions and Write out an equation using an indefinite integral and each of these functions.

(16) Find an antiderivative for the functions and Write out an equation using an indefinite integral and each of these functions.

(17)  Find an antiderivative for the functions and Write out an equation using an indefinite integral and each of these functions.

(18) Find the most general antiderivative or indefinite integral for then check you answer by differentiation.

(19) Solve the initial value problem and  

(20) Use finite approximations to estimate the area under the graph of between and

(21) Use finite approximations to estimate the area under the graph of between and

(22) Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule) estimate the area under the graph of the function between and using first two and then four rectangles.

(23) An object is shot straight upward from sea level with an initial velocity of (a) Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use for the gravitational acceleration. (b) Find a lower estimate for the height attained after 5 sec.  

(24) Use a finite sum to estimate the average of on the closed bounded interval by partitioning the interval into four subintervals of equal length and evaluating at the subinterval midpoints.

(25) Write the sum without sigma notation and then evaluate the sum.

(26) Write the sum without sigma notation and then evaluate the sum.

(27) Rewrite the expression in sigma notation.

(28) Which formula is not equivalent to the other two?

(29) Rewrite the sum using sigma notation

(30) Rewrite the sum using sigma notation

(31) Evaluate the sum

(32) Graph the function over the interval Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum given that is the (a) left-hand endpoint, (b) right-hand endpoint, (c) the midpoint of the -th subinterval . Make a separate sketch for each set of rectangles.  

(33) Express the limits as definite integrals where is a partition of

(34) Express the limits as definite integrals where is a partition of

(35) Suppose that Find,

(36) Graph the integrand and use areas to evaluate the integral

(37) Graph the integrand and use areas to evaluate the integral

(38) Graph the function and find its average value over the closed bounded interval   

(39) Evaluate the definite integral

(40) Evaluate the definite integral

(41) Find the derivative by (a) evaluating the integral and then differentiating (b) by differentiating the integral directly.

(42) Find for

(43) Find the total area between the region bounded by the curve , and the -axis, and the interval

(44) Evaluate the indefinite integral

(45) Evaluate the indefinite integral

(46) Evaluate the indefinite integral

(47) Evaluate the indefinite integral

(48) Evaluate the indefinite integral

(49) Solve the initial value problem

(50) Use the Substitution Formula to evaluate the definite integrals and

(51) Use the Substitution Formula to evaluate the definite integral

(52) Determine the area bounded by the curves and the vertical lines and

(53) Find the area of the region enclosed by the curves and

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