## Tangent Questions and Answers

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Consider x =h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (5,\ 5,\ 4) given that \left. \frac{\partial h}{\partial y}\right|_ {(5,4)}=4 \text{ and } \left. \frac{\part

Find the equation of the tangent line to the graph of the function f(x) = \sqrt{\arccos(x-1)} at the point (1, 2\pi)

a. Find the slope of this equation y = x^3+1 at the point P(-2, -7) by finding the limiting value of the slope of the secants through P. b. Find an equation of the tangent line to the curve at P(-2, -7)

Find k such that the line is tangent to the graph of the function. Function f of x is equal to k square root x Line y equal to 4 of x plus four

Find the equation of the tangent to the curve y = \sin x at x = \frac{\pi}{3}

Find: a. The slope of the curve at the given point P, b. An equation of the tangent line at P. y=-3-6x^2; \ P(-5,-153)

Find the slope m of the tangent to the curve y = 3 + 4x^2 - 2x^3 at the point where x = a.

For the given parametric curve, compute the following questions: x(t) = 5\cos t - 4 y(t) = -2\sin t + 3 0 \leq t \leq 2\pi a. Express the curve with an equation which relates x and y. b. Find the slope of the tangent line to the curve at the point t = \

a) Prove, working directly from the definition, that if f(x) = 1/x, then f'(a) = -1/a^2, for a \neq 0 . b) Prove that the tangent line to the graph of f at (a, 1/a) does not intersect the graph of f, except at (a, 1/a) .

Find an equation of the line that is tangent to the graph of f(x) = (x^2-4)^5 (2x-4)^3 for x = 1

Find the equation of a line that is parallel to line y=5x and contains the point (1,-1).

For the function f(x) = \frac{1}{\sqrt x} , find an equation of the tangent line the graph of the above function at (4, \frac{1}{2})

What is the slope of the tangent line to the polar curve r=2-\cos (\theta) at \theta = \frac {\pi}{4}?

Find an equation of the tangent line to the given curve at the given point. y = 3x^4 - x^3 - x^2 + 1, (1, 2)

Find the equation in x and y for the line tangent to the curve in polar coordinates r = 2 at the value \theta = \frac{\pi}{3}.

Find the equation in x and y for the line tangent to the curve in polar coordinates r = 3 - 6 \sin\theta at the value \theta = 0.

Find the equation in x and y for the line tangent to the curve in polar coordinates r = 2 \cos 2\theta at the value \theta = \frac{\pi}{2}.

The derivative of f(x)=x^2-4x+5 is f'(x)=2x-4. Find the equation of the tangent line to the graph of f(x) at the point (1,2)

Find the tangent line approximation T to the graph f at the given point: f(x) = \sqrt x; (2, \sqrt 2)

Find an equation of the tangent plane to the given parametric surface at the specified point. x = u^{2} + 1, y = v^{3} + 1, z = u + v; (5,2,3)

Find all of the values of t in (0,pi) for which the tangent line to the graph of x(t) = t + cos(2t), y(t) = t-cos(2t) is horizontal.
1. t = 7pi/12, 11pi/12
2. t = 5pi/12, 7pi/12
3. t = pi/3, 2pi/3
4. t = pi/12, 5pi/12
5. t = pi/6, 5pi/6
6. t = pi/4, 3pi/4

Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through (-3, -2) and (3,6)

Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Slope = -frac{4}{5}, passing through (4, -9)

Write the point-slope form of the line satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through (5, 15) and (6,18)

Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Slope = frac{1}{4} , passing through the origin

Find the equation of the tangent at the given value of x . a. y = x^2 + x -3, x = 4 b. y = 2x^2 - 7, x = -2

Let f ( x ) = x ? 1 ? x a) Find the equation of tangent line of the function f at the point 4 , f ( 4 ) . b) Use the differentials to estimate the value of f ( 4.02 )

If f(2) = -3 , f '(2) = 3/ 4 , and g(x) = f ^{-1}(x) , what is the equation of the tangent line to g(x) at x = -3 ?

Find an equation of the tangent plane at the given point. g(x,y) = e^{x/y}; (2,1)

Find an equation of the tangent plane at the given point. F(r,s) = r^2s^{-1/2} + s^{-3}; (2,1)

Find an equation of the tangent plane at the given point. G(u,w) = \sin(uw), (pi/6, 1)

Find all the values in the interval |-2 \pi, 2 \pi| at which the graph of f(x)=x- \cos (x) has a horizontal tangent line.

Determine the slope of the tangent line to the curve y=\frac{4}{x} at the point (3,1.3333). The equation of this tangent line can be written in the form y=mx+b

Let f(x)=3x^4-x^2+7 a. Find f'(x) b. Find the slope of the tangent line to the graph of f at x=3 c. Find an equation of the line tangent to the graph of f at =3

Determine the slope from the tangent line to the curve y=3x^3 at the point (-4,-192). The equation of this tangent line can be written in the form y=mx+b.

Find the points on the graph f(x,y) = (x + 1)y^2 at which the tangent plane is horizontal.

Find the points on the graph f(x,y) = 3x^2 - xy - y^2 at which the tangent plane is horizontal.

Find the points on the graph of z = xy^3 + 8y^-1 where the tangent plane is parallel to 2x + 7y + 2z = 0 .

Find a point on the graph f(x) = x^2 + x + 3 between x = 1 and x = 2, where the tangent line is parallel to the line connecting (1, 5) and (2,9).

Which of the following is an approximate equation of the line tangent to the graph of f(x) = x^4 + 2x^2 at the point where f'(x) = 1? A. y \approx x - 0.122 B. y \approx x - 2.146 C. y \approx x + 0.763 D. y \approx x + 3.281

Find an equation of the tangent plane to the surface z=-x^2+2y^2+2x+2y-2 at the points (3, 3, 19).

Consider the function f(x) = xe^x - x^2} 1) Find the tangent line to the graph of y = f(x) when x = 0. 2) Find the points where the graph of y = f(x) has a horizontal tangent. 3) Graph the function f(x), the tangent line found in (1) and the points you f

Find the equation of the line or lines tangent to the graph where indicated: a. y = e^{2x - 3}, when x = 3/2 b. y = e^{-x^2}, at inflection point(s) c. y = xe^{-x}, at inflection point(s) d. y = x \ln x, wen x = 1 e. y = \ln x^2, when x = 2 f. y = x^2 + 2

Find the equation of the line tangent to x^3+x+5x+x^2=16 at the point (1,2)

Consider the following. f(x)=\tan^{2} x \quad \Big( -\frac{\pi}{4}, 1 \Big) Find an equation of the tangent line to the graph of f at the given point.

Determine the points in the interval [0, 2\pi] at which the graph of the function below has a horizontal tangent. f(x)=-14\cos x +7\sin 2x

Find a vector parametrization equation \vec{r}(t) for the line through the points P=(0, 0, 5) and Q=(-2, -1, 2) for each of the given conditions on the parameter t. If the points P and Q correspond to the parameter values t=0 and t=-2, respectively.

Find the parametric equations for the tangent line to the curve x=t^4-1, y=t^2+1, z=t^3 at the point (15, 5, 8).

Let g be a function that is defined for all x, x \neq 2, such that g(3) = 4 and the derivative of g is g'(x) = \frac{x^2 - 16}{x - 2}\ with\ x \neq 2. Write an equation for the tangent line to the graph of g at the point x= 3. Does this tangent line lie

Find an equation for the line tangent to the curve at the point defined by the given value of t x = 6 t 2 ? 3 , y = t 3 , t = 1 A) 5 12 x + 1 4 B) 5 12 x ? 1 4 C) 5 6 x + 1 D) 12 5 x + 1 4

Find the equation of the tangent plane to the surface determined by x^2y^4 + z - 30 = 0 at x = 3, y =5

Consider the function f(x, y, z)= 3x^2\sqrt{y} + \cos(3xz) - 9. Find the equation of the tangent plane at the point P(1, 9, \frac{\pi}{6}) for the function.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y^2(y^2-4)=x^2(x^2-5) \quad (0,-2)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 \quad (-3\sqrt{3}, 1)

Find an equation for the tangent plane to the surface given below at the indicated point. * The graph of z=f(x,y) implicitly defined by xy+xz+yz \ln \left(z^2 + 1 \right) at the point \left( 2, \ 2, \ 0 \right) . .

Find an equation for the tangent plane to the surface given below at the indicated point. * The graph of z=f(x,y) = x^2 y^3 at the point on the surface corresponding to x=2 \text{ and} y=1.

Given the equation 48x^6+9x^{48}y |y=58, find \frac {dy}{dx}. Also, find the tangent line to the curve at (1,1). Write your answer in mx+b format.

Consider the curve given by the equation (x^2 + y^3)^2 = 4xy Use implicit differentiation to find the equation of the tangent line at the point (1,1)

Find the equation if the line tangent to the following curve at x=1. Write your answer in y=mx+b format.

Find the equation of the line tangent to the following curve at x = 1. Give your answer in y = mx + b format. y = 2x^2 + 6x - 4.