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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Find the derivatives of the function.y = (4x + 3)4(x + 1)-3
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.y2 - 2x - 4y - 1 = 0, (-2, 1)
Find the derivatives of the function.y = (2x - 5)-1(x2 - 5x)6
Find the derivative of y with respect to x, t, or θ, as appropriate. (x² + 1)5 (VT-x V1 - x y = In
Find the derivative of y with respect to x, t, or θ, as appropriate. y = ln Vsin 0 cos 0 1 + 2 ln 0
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.6x2 + 3xy + 2y2 + 17y - 6 = 0, (-1, 0)
Find the derivatives of the function.y = xe-x + ex3
Find the derivatives of the function. k(x) = x² sec X
Find the derivatives of the function.y = (1 + 2x)e-2x
Find the derivative of y with respect to x, t, or θ, as appropriate. y = ln (x + 1)5 √ (x + 2)20
Find the derivative of y with respect to x, t, or θ, as appropriate.y = ln (sec (ln θ))
Find the derivatives of the function. f(x) = √7 + x sec x
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.2xy + π sin y = 2π, (1,π/2)
Find the derivatives of the function.y = (x2 - 2x + 2)e5x/2
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.x sin 2y = y cos 2x, (π/4, π/2)
Find the derivatives of the function.y = (9x2 - 6x + 2)ex3
Find equations for the tangent and normal to the cissoid of Diocles y2(2 - x) = x3 at (1, 1). y 1 0 3 y²(2-x) = x³ 1 (1, 1)
Find the derivatives of the function. g(x) = tan 3x (x + 7)4
Find the derivatives of the function. g(t) = 1 + sin 3t 3 - 2t
Find the slopes of the curve y4 = y2 - x2 at the two points shown here. y 0 ( 大义 |y'=y2-x2
Find the slopes of the devil’s curve y4 - 4y2 = x4 - 9x2 at the four indicated points. (-3, 2) -3 (-3,-2) 2 y -2 y4 - 4y² = x49x² (3, 2) X 3 (3,-2)
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.y = 2sin(px - y), (1, 0)
Find the derivatives of the function. f(0) = G sin Ꮎ 2 6)²³ 1 + cos 0
Find the derivatives of the function.h(x) = x tan (√1x) + 7
Verify that the given point is on the curve and find the lines that are (a) Tangent (b) Normal to the curve at the given point.x2 cos2 y - sin y = 0, (0, π)
Find the two points where the curve x2 + xy + y2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents?
a. Find the slope of the folium of Descartes x3 + y3 - 9xy = 0 at the points (4, 2) and (2, 4).b. At what point other than the origin does the folium have a horizontal tangent?c. Find the coordinates
Find the derivatives of the function. sec Vetan r = sec 1 0
Find the normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0.
Find the derivatives of the function. 9 = sin t Vt + 1
Let p and q be integers with q > 0. If y = xp/q, differentiate the equivalent equation yq = xp implicitly and show that, for y ≠ 0, d dx b/dx- = x(P/9)-1¸
Find the derivatives of the function. 9 cot sin t t
Show that if it is possible to draw three normals from the point (a, 0) to the parabola x = y2 shown in the accompanying diagram, then a must be greater than 1/2. One of the normals is the x-axis.
Find the derivatives of the function. y cos (e-0²) =
Is there anything special about the tangents to the curves y2 = x3 and 2x2 + 3y2 = 5 at the points (1, ±1)? Give reasons for your answer. y 2x² + 3y² = 5 0 y² = x³ (1, 1) (1, -1) X
Find the derivatives of the function.r = sin (θ2)cos (2θ)
The line that is normal to the curve x2 + 2xy - 3y2 = 0 at (1, 1) intersects the curve at what other point?
Find the derivatives of the function.y = θ3e-2θ cos 5θ
Find dy/dt.y = sin2 (πt - 2)
Find dy/dt. = y 13 1² 4t 3
Verify that the following pairs of curves meet orthogonally.a. x2 + y2 = 4, x2 = 3y2b. x = 1 - y2, x = 1/3y2
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find dy/dt.y = sec2πt
Find dy/dt. y = 3t - 4 5t + 2 -5
The graph of y2 = x3 is called a semicubical parabola and is shown in the accompanying figure. Determine the constant b so that the line y = -1/3x + b meets this graph orthogonally.
Find dy/dt.y = (1 + cos 2t)-4
Find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related? Explain the relationship
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find dy/dt.y = (1 + cot (t/2))-2
Find dy/dt. y (5 sin (3)) = cos 5 sin
Find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related? Explain the relationship
Find dy/dt.y = (t tan t)10
Find dy/dt. 3 t ; = (1 + tan ¹ (1/2)) ² y
Find dy/dt.y = (t-3/4 sin t)4/3
Find dy/dt.y = ecos2 (πt-1)
Find dy/dt. y 1 6 + cos²(71))3
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find dy/dt.y = (esin (t/2))3
Find dy/dt. y = V1 + cos (1²)
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find dy/dt. y = = 3t (2t² - 5)4
Find dy/dt. y = 4 sin (V1 + Vt)
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find dy/dt.y = sin (cos (2t - 5))
Find dy/dt. y = tan² (sin³ t)
Find dy/dt. y cos4 (sec² 3t)
Find dy/dt. y = 3t+ V2 + V1 - t
Find y″. y = (1 - √x)-¹ X
Find y″. y = + co cot(3x – 1)
Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a
Find y″. y = 9 tan X 3
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) = u5 + 1, u = g(x) = √x, x = 1
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) = 1 - u' И = g(x) = 1 1 - x' x = -1
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) = cot TTU 10' u = g(x) = 5√x, x = 1
Suppose that the functions ƒ and g and their derivatives with respect to x have the following values at x = 0 and x = 1.Find the derivatives with respect to x of the following combinations at the
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) = 2u u² + 1 u = g(x) = 10x² + x + 1, x = 0
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) = u + 1 cos² u u = g(x) = TTX, x = 1/4
Find the value of (ƒ ∘ g)′ at the given value of x. f(u) 2 = (a + 1)². u = n g(x) = X² - 1, x = -1
Find y″.y = x (2x + 1)4
Find y″.y = x2 (x3 - 1)5
Find y″.y = ex2 + 5x
The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x isand is
Find y″.y = sin (x2ex)
Find the tangent to y = ((x - 1)/(x + 1))² at x = 0.
Find the tangent to Vx²x + 7 at x = 2. y = √x²
Assume that ƒ′(3) = -1, g′(2) = 5, g(2) = 3, and y = ƒ(g(x)). What is y′ at x = 2?
If r = sin (ƒ(t)), ƒ(0) = π/3, and ƒ′(0) = 4, then what is dr/dt at t = 0?
For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equationwhere g is the constant
Find ds/dt when θ = 3π/2 if s = cos θ and dθ/dt = 5.
Find dy/dt when x = 1 if y = x2 + 7x - 5 and dx/dt = 1/3.
What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the function.Find dy/dx if y = x by
Graph the function y = 2cos2x for -2 ≤ x ≤ 3.5. Then, on the same screen, graphfor h = 1.0, 0.5, and 0.2. Experiment with other values of h, including negative values. What do you see happening
What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the function.Find dy/dx if y = x3/2 by
Suppose that ƒ(x) = x2 and g(x) = |x|. Then the compositesare both differentiable at x = 0 even though g itself is not differentiable at x = 0. Does this contradict the Chain Rule? Explain. (fog)(x)
a. Find the tangent to the curve y = 2 tan(πx/4) at x = 1.b. What is the smallest value the slope of the curve can ever have on the interval -2 < x < 2? Give reasons for your answer.
Graph y = -2x sin (x2) for -2 ≤ x ≤ 3. Then, on the same screen, graphfor h = 1.0, 0.7, and 0.3. Experiment with other values of h. What do you see happening as h → 0? Explain this behavior. y
a. Find equations for the tangents to the curves y = sin 2x and y = -sin (x/2) at the origin. Is there anything special about how the tangents are related? Give reasons for your answer.b. Can
Using the Chain Rule, show that the Power Rule (d/dx)xn = nxn-1 holds for the functions xn. x1/4 = √√x X
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