# Calculus AB

posted by Annie

Sorry but I've got a lot of problems that I don't understand.

1) Let f(x)= (3x-1)e^x. For which value of x is the slope of the tangent line to f positive? Negative? Zero?

2) Find an equation of the tangent line to the oven curve at the specified point. Sketch the curve and the tangent line to check your answer.
a) y= x^2 + (e^x)/(x^2 +1) at the point x=3.
b) y= 2x(e^x) at the point x=0

3) Suppose that the tangent line to the graph of f at (2,f(2)) is y=3x+4. Find the tangent line to the graph of xf(x).

4) The line that is normal to the curve x^2 + 2xy - 3y^2=0 at (1,1) intersects at what other point?

5) Evaluate (d^2)/(dx^2) * (1)/(1-2x)

6) Find dy/dx in terms of x and y.
a) x^3 +y^3=3xy^2
b) cos(xy^2)=y

7) Find ((d^2)y)/(dx^2) in terms of x and y.
a) 2x^2 -3y^2=4
b) y+ siny=x

I know it's a lot but I just really don't understand how to do them.

1. Steve

#1 f' = (3x+2)e^x
since e^x is always positive, just determine where 3x+2 is positive, etc.

#2 remember the point-slope form of a line.

a) y= x^2 + (e^x)/(x^2 +1) at the point x=3.
y' = 2x + (x-1)^2/(x+1)^2 e^x
y(3) = 9+e^3/10
y'(3) = 6+e^3/4
So, you want the line
y-(9+e^3/10) = (6+e^3/4)(x-3)

b) y= 2x(e^x) at the point x=0
y' = 2(x+1)e^x
y(0) = 0
y'(0) = 2
The tangent line is thus
y = 2x

3) d/dx (x*f) = f + xf'
f'(2) = 3 since that is the slope of the tangent line.
f(2) = 10 since the line meets it there.
at x=2, x*f has slope f(2) + 2*2 = f(2)+4 = 14
So the tangent line there is
y-20 = 14(x-2)

4)x^2 + 2xy - 3y^2=0
This is just a pair of intersecting lines (a degenerate conic)
The lines are y=x and y = -x/3
At (1,1) the normal line is
y-1 = -(x-1)
y = -x+2
That line meets y = -x/3
(3,-1)

5) Not quite sure what you mean, but if
y = 1/(1-2x)
y' = 2/(1-2x)^2
y" = 8/(1-2x)^3

For 6 and 7, just remember the chain rule and the product rule. When there are x's and y's, there will be y' floating around.

6)
(a) x^3 + y^3 = 3xy^2
3x^2 + 3y^2 y' = 3y^2 + 6xy y'
y'(3y^2-6xy) = 3y^2-3x^2
y' = (x^2-y^2)/(y^2-2xy)

(b) cos(xy^2) = y
-sin(xy^2)(y^2+2xyy') = y'
y' = -(y^2 sin(xy^2))/(1+2xy sin(xy^2))

7)
(a) 2x^2-3y^2 = 4
4x-6yy' = 0
y' = 2x/3y
y" = (2*3y - 3x*3y')/9y^2
= (6y-9x(2x/3y))/9y^2
= 2(1-x^2)/3y

(b) y+siny = x
y' + cosy y' = 1
y' = 1/(1+cosy)
y" = siny/(1+cosy)^2 y'
= siny/(1+cosy)^3

2. Annie

Thank you so much. I kind of understand the chain rule and product rule, but it's always hard for me to determine how to do/use them.

## Similar Questions

1. ### Math (Calculus 121)

The derivative tells you the slope at any point. df/dx = 8 x^3 - 12 x When x = 3, the slope is m = 8*27 - 36 = 180 and the equation of the tangent line is (y - 228) = 180*(x - 3) y = 180 x - 540 + 228 = 180 x - 312 wow, that was absolutely …
2. ### calculus

First I would find the first derivative which is the slope and then would I plug in the given value into the original equation to find y and then use y-y_1=m(x-x_1)?
3. ### calculus

Consider line segments which are tangent to a point on the right half (x>0) of the curve y = x^2 + 1 and connect the tangent point to the x-axis. If the tangent point is close to the y-axis, the line segment is long. If the tangent …
4. ### Math

I have a series of questions that I did. They lead up to the last question I can't solve. Could you check my math and help me with the last question?
5. ### Calculus: for mathmate

Find the slope of the tangent line to the graph of the given function at any point: 1. f(x)= 13 I understand that you said the tangent to the line is the coefficient of x, but what would be the slope of this problem. For problems like …
6. ### College Calculus

Let f(x) = 3x^2 - 2x + 1 a) write an equation of the line going through the points (1, f(1)), and (2, f(2)). b) Find a point on the graph of f where the tangent line to the graph has the same slope as the line in part (a). Write the …
7. ### Calculus

There are two neighboring points P and Q on the curve f(x)=x^2 (a) Determine m(h), the slope of the straight line PQ (b) The slope, S, of the tangent at P is defined as S=lim h->0 m(h). Find the value of S. (c) Use this value of …
8. ### Calculus

Consider the curve given by y^2 = 2+xy (a) show that dy/dx= y/(2y-x) (b) Find all points (x,y) on the curve where the line tangent to the curve has slope 1/2. (c) Show that there are now points (x,y) on the curve where the line tangent …
9. ### calculus

Describe the slope of the tangent line to the curve defined by x 2 + xy - y 2 = 5 when x = 2. positive negative zero undefined
10. ### slope

Find the slope of the line passing through the given points using the slope formula. Enter the slope in simplest form. Describe the slope as positive, negative, zero, or undefined. (8,2) and (15,9) my answer = 1 the slope is posistive

More Similar Questions