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EVALUATING INTEGRALS.

Evaluate the integral shown below. (Hint: Try the substitution u = (7×2 + 3).) 1) a x dx (7×2 + 3)5 Evaluate the integral shown below. (Hint: Apply a property of logarithms first.) 2) a l n x6 x dx 1 Use the Fundamental Theorem of Calculus to find the derivative shown below. 3) a d d x x5 0 sin t dt For the function shown below, sketch a graph of the function, and then find the SMALLEST possible value and the LARGEST possible value for a Riemann sum of the function on the given interval as instructed. 4) f(x) = x2 ; between x = 3 and x = 7 with four rectangles of equal width. ^ CHARACTERISTICS and BEHAVIOR OF FUNCTIONS. Use l’Hopital’s rule to find the limit below. 5) lim x Q 5x + 9 6×2 + 3x – 9 ^ Use l’Hopital’s rule to find the limit below. (Hint: The indeterminate form is f(x)g(x). ) 6) lim x ’ 1 + 2 x3 x 2 Solve the following problem. 7) The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall. 9′ wall 30′ Hint: Let “h” be the height on the building where the ladder touches; let “x” be the distance on the ground between the wall and the foot of the ladder. Use similar triangles and the Pythagorean Theorem to write the length of the beam “L” as a function of “x”. Also note that a radical function is minimized when it radicand is minimized. For the function shown below, identify its local and absolute extreme values (if any), saying where they occur. 8) f(x) = -x3- 9×2 – 24x + 3 3 Find a value for “c” that satisfies the equation ƍ f(b) – f(a) b – a = f (c) in the conclusion of the Mean Value Theorem for the function and interval shown below. 9) f(x) = x + 75 x , on the interval [3, 25] DERIVATIVES. Find the equation of the tangent line to the curve whose function is shown below at the given point. 10) x5y5 = 32, tangent at (2, 1) Use implicit differentiation to find dy/dx. 11) xy + x + y = x2y2 Given y = f(u) and u = g(x), find dy/dx = fƍ(g(x))gƍ(x). 12) y = u(u – 1), u = x2 + x 4 Find y ƍ. 13) y = (4x – 5) (4×3 – x2 + 1) Find the derivative of the function “y” shown below. 14) y = x2 + 8x + 3 x Solve the problem below. 15) One airplane is approaching an airport from the north at 163 km/hr. A second airplane approaches from the east at 261 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 31 km away from the airport and the westbound plane is 18 km from the airport. FUNCTIONS, LIMITS and CONTINUITY. Find the intervals on which the function shown below is continuous. 16) y = x + 2 x2 – 8x + 7 5 A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number Ή is given. Find a number Έ > 0 such that for all x, 0 < x – c < Έ f(x) – L < Ή. 17) f(x) = 10x – 1, L = 29, c = 3, and Ή = 0.01 Find all points “x” where the function shown below is discontinuous. 18) Solve the “composite function” problem shown below. 19) If f(x) = x + 4 and g(x) = 8x – 8, find f(g(x)). What is f(g(0))? Find the limit shown below, if it exists. 20) lim x 5 x2 – 25 x2 – 6x + 5 6