Calculus 1: answer 45 easy Qs in two days

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Yousef Almarshdi MATH125­050, section 050, Summer 1 2017 Instructor: James Wang WebAssign Sec. 2.4 HW (Homework) Current Score : – / 10 Due : Wednesday, June 7 2017 11:59 PM CDT 1. –/1 pointsSEssCalcET2 2.4.005.MI. Differentiate. F(y) = 1 y2 − 3 y4 (y + 9y3) 2. –/1 pointsSEssCalcET2 2.4.007. Differentiate. f(x) = sin x + f '(x) = 7 cot x 8 3. –/1 pointsSEssCalcET2 2.4.031. (a) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point 1, 1 . 2 y= (b) Illustrate part (a) by graphing the curve and tangent line on the same screen. 4. –/1 pointsSEssCalcET2 2.4.033. If f(x) = x2/(4 + x), find f ''(2). f ''(2) = 5. –/1 pointsSEssCalcET2 2.4.040. Suppose f(π/3) = 5 and f '(π/3) = −3, and let g(x) = f(x) sin x and h(x) = (cos x)/f(x). Find the following. (a) g'(π/3) (b) h'(π/3) 6. –/1 pointsSEssCalcET2 2.4.041. Suppose that f(5) = 1, f '(5) = 3, g(5) = −5, and g'(5) = 4. Find the following values. (a) (fg)'(5) (b) (f/g)'(5) (c) (g/f)'(5) 7. –/1 pointsSEssCalcET2 2.4.043. If f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x). (a) Find u'(1). (b) Find v'(5). 8. –/1 pointsSEssCalcET2 2.4.044. Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown. (a) Find P'(2). (b) Find Q'(7). 9. –/1 pointsSEssCalcET2 2.4.046. If f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x4f(x) y' = x3f '(x) − 3x4f(x) y' = x4f(x) − 4x3f '(x) y' = x4f '(x) + 4x3f(x) y' = x4f(x) + 4x3f '(x) y' = x3f '(x) + 3x4f(x) (b) y= f(x) x6 xf '(x) + 5f(x) y' = x6 xf '(x) − 6f(x) y' = x7 xf(x) + 6f '(x) y' = x6 xf(x) − 6f '(x) y' = x5 xf '(x) − 5f(x) y' = x7 (c) 8 y= x f(x) f '(x)(7x7) − x8f '(x) y' = [f(x)]2 f '(x)(7x8) + x7f(x) y' = [f(x)]2 f '(x)(8x7) − x8f(x) y' = [f(x)]2 f(x)(7x8) + x8f '(x) y' = [f(x)]2 f(x)(8x7) − x8f '(x) y' = [f(x)]2 (d) y= 3 + xf(x) x xf(x) + 2x2f '(x) − 3 y' = 2x3/2 xf(x) − 3x2f '(x) + 3 y' = 2x3/2 xf '(x) − 2x2f '(x) + 3 y' = 2x3/2 xf '(x) + 3x2f(x) − 2 y' = 2x3/2 xf(x) + 3x2f(x) − 2 y' = 2x3/2 10.–/1 pointsSEssCalcET2 2.4.AE.004. Video Example EXAMPLE 4 y= x2 Let +x−1 x3 + 9 . Then (x3 + 9) d dx y' = − (x2 + x − 1) d dx (x3 + 9)2 (x3 + 9) = − (x2 + x − 1) (x3 + 9)2 (x3 + 9) = − (x3 + 9)2 . = (x3 + 9)2 Yousef Almarshdi MATH125­050, section 050, Summer 1 2017 Instructor: James Wang WebAssign Sec. 2.5 HW (Homework) Current Score : – / 12 Due : Wednesday, June 7 2017 11:59 PM CDT 1. –/1 pointsSEssCalcET2 2.5.013. Find the derivative of the function. y = cos(a9 + x9) y'(x) = 2. –/1 pointsSEssCalcET2 2.5.032. Find the derivative of the function. y = x sin y'(x) = 3 x 3. –/1 pointsSEssCalcET2 2.5.036. Find the derivative of the function. y = cos(cos(cos x)) y' = 4. –/1 pointsSEssCalcET2 2.5.038. Find the derivative of the function. y= y' = 7x + 7x + 7x 5. –/1 pointsSEssCalcET2 2.5.053. If F(x) = f(g(x)), where f(3) = 3, f '(3) = 6, f '(−2) = 4, g(−2) = 3, and g'(−2) = 2, find F '(−2). F '(−2) = 6. –/1 pointsSEssCalcET2 2.5.054. If h(x) = 3 + 2f(x) , where f(2) = 3 and f '(2) = 4, find h'(2). h'(2) = 7. –/1 pointsSEssCalcET2 2.5.055. A table of values for f, g, f ', and g' is given. x f(x) g(x) f '(x) g'(x) 1 3 2 4 6 2 1 8 5 7 3 7 2 7 9 (a) If h(x) = f(g(x)), find h'(1). h'(1) = (b) If H(x) = g(f(x)), find H'(2). H'(2) = 8. –/1 pointsSEssCalcET2 2.5.056. Let f and g be the functions in the table below. x f(x) g(x) f '(x) g'(x) 1 3 2 4 6 2 1 3 5 7 3 2 1 7 9 (a) If F(x) = f(f(x)), find F '(3). F '(3) = (b) If G(x) = g(g(x)), find G'(1). G'(1) = 9. –/1 pointsSEssCalcET2 2.5.057. If f and g are the functions whose graphs are shown, let u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why. (If an answer does not exist, enter DNE.) (a) u'(1) = It does exist. u'(1) does not exist because f '(1) does not exist. u'(1) does not exist because g'(1) does not exist. u'(1) does not exist because f '(3) does not exist. u'(1) does not exist because g'(2) does not exist. (b) v'(1) = It does exist. v'(1) does not exist because f '(1) does not exist. v'(1) does not exist because g'(1) does not exist. v'(1) does not exist because f '(3) does not exist. v'(1) does not exist because g'(2) does not exist. (c) w'(1) = It does exist. w'(1) does not exist because f '(1) does not exist. w'(1) does not exist because g'(1) does not exist. w'(1) does not exist because f '(3) does not exist. w'(1) does not exist because g'(2) does not exist. 10.–/1 pointsSEssCalcET2 2.5.061. Let r(x) = f(g(h(x))), where h(1) = 5, g(5) = 4, h'(1) = 4, g'(5) = 3, and f '(4) = 6. Find r'(1). r'(1) = 11.–/1 pointsSEssCalcET2 2.5.063. Find the 30th derivative of y = cos 3x. f (30)(x) = 12.–/1 pointsSEssCalcET2 2.5.AE.002. Video Example Differentiate (a) y = sin(x6) and (b) sin6 x. EXAMPLE 2 SOLUTION (a) If y = sin(x6), then the outer function is the sine function and the inner function is the power function, so the Chain Rule gives dy d = dx dx sin (x6) outer function evaluated at inner function = cos (x6) derivative of outer function evaluated at inner function · = . (b) Note that sin6 x = (sin x) . Here the outer function is the power function and the inner function is the sine function. So dy d = (sin x) dx dx inner function = . 6 = 6 · (sin x)5 derivative of outer function evaluated at inner function · derivative of inner function derivative of inner function WebAssign Sec. 2.6 HW (Homework) Current Score : – / 6 Due : Wednesday, June 7 2017 11:59 PM CDT 1. –/1 pointsSEssCalcET2 2.6.009. Find dy/dx by implicit differentiation. 5 cos x sin y = 1 y' = 2. –/1 pointsSEssCalcET2 2.6.013. Find dy/dx by implicit differentiation. xy = 3 + x2y dy = dx 3. –/1 pointsSEssCalcET2 2.6.017. If f(x) + x2[f(x)]4 = 18 and f(1) = 2, find f '(1). f '(1) = Yousef Almarshdi MATH125­050, section 050, Summer 1 2017 Instructor: James Wang 4. –/1 pointsSEssCalcET2 2.6.020. Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 2xy − y2 + x = 5, (3, 7) (hyperbola) y= 5. –/1 pointsSEssCalcET2 2.6.022. Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 (−3 3 , 1) (astroid) y= 6. –/1 pointsSEssCalcET2 2.6.046. Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3). y = (smaller slope) y = (larger slope) WebAssign Sec. 2.7 HW (Homework) Current Score : – / 9 Yousef Almarshdi MATH125­050, section 050, Summer 1 2017 Instructor: James Wang Due : Thursday, June 8 2017 11:59 PM CDT 1. –/1 pointsSEssCalcET2 2.7.003. Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 36 cm2? cm2/s 2. –/1 pointsSEssCalcET2 2.7.011. If a snowball melts so that its surface area decreases at a rate of 7 cm2/min, find the rate at which the diameter decreases when the diameter is 12 cm. cm/min 3. –/1 pointsSEssCalcET2 2.7.012. At noon, ship A is 180 km west of ship B. Ship A is sailing east at 30 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM? km/h 4. –/1 pointsSEssCalcET2 2.7.013. A plane flying horizontally at an altitude of 1 mi and a speed of 420 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. (Round your answer to the nearest whole number.) mi/h 5. –/1 pointsSEssCalcET2 2.7.014. A street light is mounted at the top of a 15­ft­tall pole. A man 6 ft tall walks away from the pole with a speed of 7 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole? ft/s 6. –/1 pointsSEssCalcET2 2.7.015.MI. Two cars start moving from the same point. One travels south at 48 mi/h and the other travels west at 20 mi/h. At what rate is the distance between the cars increasing four hours later? mi/h 7. –/1 pointsSEssCalcET2 2.7.016.MI. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.5 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? (Round your answer to one decimal place.) m/s 8. –/1 pointsSEssCalcET2 2.7.017. A man starts walking north at 3 ft/s from a point P. Five minutes later a woman starts walking south at 7 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking? (Round your answer to two decimal places.) ft/s 9. –/1 pointsSEssCalcET2 2.7.021. At noon, ship A is 90 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 5 km/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.) km/h 1. –/1 pointsSEssCalcET2 2.8.005. Find the linear approximation of the function f(x) = 4 − x at a = 0. L(x) = Use L(x) to approximate the numbers 3.9 and 3.9 ≈ 3.99 ≈ Illustrate by graphing f and the tangent line. 3.99 . (Round your answers to four decimal places.) 2. –/1 pointsSEssCalcET2 2.8.006.MI. Find the linear approximation of the function g(x) = 5 1 + x at a = 0. g(x) ≈ Use it to approximate the numbers 5 5 0.95 and 0.95 ≈ 5 1.1 ≈ Illustrate by graphing g and the tangent line. 5 1.1 . (Round your answers to three decimal places.) 3. –/1 pointsSEssCalcET2 2.8.013. Use a linear approximation (or differentials) to estimate the given number. (8.03)2/3 4. –/1 pointsSEssCalcET2 2.8.019. Ley y = tan x. (a) Find the differential dy. dy = (b) Evaluate dy and Δy if x = π/4 and dx = −0.2. (Round Δy to four decimal places.) dy = Δy = 5. –/1 pointsSEssCalcET2 2.8.024. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with diameter 56 m. (Round your answer to two decimal places.) m3 6. –/1 pointsSEssCalcET2 2.8.029. Suppose that the only information we have about a function f is that f(1) = 7 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f(0.99) and f(1.01). f(0.99) ≈ f(1.01) ≈ (b) Are your estimates in part (a) too large or too small? Explain. The slopes of the tangent lines are positive, but the tangents are becoming less steep. So the tangent lines lie above the curve f. Thus, the estimates are too small. The slopes of the tangent lines are negative, but the tangents are becoming steeper. So the tangent lines lie below the curve f. Thus, the estimates are too small. The slopes of the tangent lines are positive, but the tangents are becoming less steep. So the tangent lines lie above the curve f. Thus, the estimates are too large. The slopes of the tangent lines are negative, but the tangents are becoming steeper. So the tangent lines lie below the curve f. Thus, the estimates are too large. 7. –/1 pointsSEssCalcET2 2.8.030. Suppose that we don't have a formula for g(x) but we know that g(2) = −4 and g'(x) = x2 + 12 for all x. (a) Use a linear approximation to estimate g(1.9) and g(2.1). g(1.9) ≈ g(2.1) ≈ (b) Are your estimates in part (a) too large or too small? Explain. The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie below the curve. Thus, the estimates are too small. The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie below the curve. Thus, the estimates are too small. The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie above the curve. Thus, the estimates are too large. The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie above the curve. Thus, the estimates are too large. 8. –/1 pointsSEssCalcET2 2.8.506.XP. Compute Δy and dy for the given values of x and dx = Δx. (Round your answers to three decimal places.) y= x , x = 1, Δx = 1 Δy = dy = Sketch a diagram showing the line segments with lengths dx, dy, and Δy.
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