MATH 1680 Flashcards
e^2x
()I=
4e
(4e)I=0
e^2x-1
(e^2x-1)I= 2e^2x-1
xe^x
(xe^x)I= (1+x)e^x
(e^x-e^-x)/2
((e^x-e^-x)/2)I= 1/2(e^x+e^-x= (e^x+e^-x)/2
e^x/x
(e^x/x)I= (x3^x-e^x)/x^2
xe^x-e^x
(xe^x-e^x)I= xe^x+e^x-e^x= xe^x
y = e^−5x6
-30x^5e^{-5x^6}
f(x) = e^x^2 + 8
2xe^{x^2+8}
g(x) = 6e√x
3e^sqrt{x}/sqrt{x}
y = x2ex − 2xex + 8ex
x^2e^x+6e^x
Find an equation of the tangent line to the graph of the function at the given point.
g(x) = ex5 − 6x, (−1, e5)
y=-e^5x
Find an equation of the tangent line to the graph of the function at the given point.
y = (e2x − 3)2, (0, 4)
y=-8x+4
Find the second derivative.
f(x) = 7e−x − 9e−7x
Find
dy
dx
implicitly.
x2y − ey − 9 = 0
-\frac{2xy}{x^2-e^y}
Find dy/dx implicitly.
exy + x2 − y2 = 15
-\frac{ye^{xy}+2x}{xe^{xy}-2y}
Find the second derivative.
f(x) = (7 + 4x)e−3x
-24e^{-3x}+36xe^{-3x}+63e^{-3x}
Differentiate.
y =
x
ex
\frac{1-x}{e^x}
Differentiate.
g(x) = 5ex√x
\frac{5e^x+10xe^x}{2\sqrt{x}}
Differentiate.
f(x) =
2 − xex
x + ex
\frac{-e^{2x}-x^2e^x-2-2e^x}{\left(x+e^x\right)^2}
Find an equation of the tangent line to the given curve at the specified point.
y =
ex
x
, (1, e)
y=e
Find the derivative of the function.
f(z) = ez/(z − 8)
e^{\left(\frac{z}{z-8}\right)}\cdot -\frac{8}{\left(z-8\right)^2}
Find the derivative of the function.
f(x) = ln(4 − x6)
\frac{1}{4-x^6}\cdot \left(-6x^5\right)
Find the derivative of the function.
y = (ln(x6))2
\frac{2\ln \left(x^6\right)^2}{\ln \left(x\right)\cdot x}
Find the derivative of the function.
y =
ln(x)
x2
\frac{1-2\ln \left(x\right)}{x^3}
Find the derivative of the function.
y = ln
√(x + 7/x − 7)
-\frac{7}{x^2-49}
Constant Rule
∫k dx = kx + C
Sum Rule
d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]
Difference Rule
∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx.
Simple Power Rule
∫xn dx = xn+1/(n + 1) + C, where n ≠ -1
Simple Log Rule
∫logax dx = x logax - x/ln a + C. ∫ln x dx = x ln x - x + C.
Simple Exponential Rule
∫ax dx = ax/ln a + C. ∫ex dx = ex + C [Because ln e = 1]
Constant Multiple Rule
∫kf(x) dx= k ∫ f(x) dx
∫-5dx= -5x+C
-5x+C or prime= -5
∫3xdx= 3 (x)^1+1/1+1
3/2 x^2 +c
USUB (3-4x^2)3 (-8x) dx
(3-4x2)4/4 + C
USUB (x3+1)1/2 (3x2) dx
(x3+1)3/2 / (3/2) + C
USUB (x-3)5/2
(x-3)7/2 / 7/2 +C
USUB (X4+3x2) (6x) dx
USUB (X4+3x2) (6x) + C
Identify u and
du
dx
for the integral
du/dx dx.
(7 − 2x2)5(−4x) dx
u= 7-2x2
du=-4x
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
(x − 6)3⁄2 dx
2/5(x-6)^5/2
Identify u and
du/dx for the integral
du/dx dx.
4x3 Square root of x4 + 3 dx
u= x4+3
du=4x3
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
5 − 4x2 cube root of (−8x) dx
\frac{3}{4}\left(5-4x^2\right)^{\left(\frac{4}{3}\right)}+C
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
x(1 − 8x2)5 dx
-1/96(1-8x2)6 +C
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
x2/(4x3 + 7)2 dx
-1/(48x3+84)+C
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
(2x3 + 4x)3(3x2 + 2) dx
2x12+16x10+48x8+64x6+32x4+C
d(x^n)/dx =
nx^n-1
∫ xn dx=
(xn+1) / (n+1) + C
Integral of ax is, ∫ ax dx
ax / ln a + C
Integral of 1/x is, ∫ 1/x dx
= ln |x| + C
Use a symbolic integration utility to find the indefinite integral. (Use C for the constant of integration.)
1
x6
e1⁄(15x5) dx
3e(1/15x5) + C
What is ∂ called?
impartial derivative
∂f / ∂x = lim h → 0 [ f(x + h, y) - f(x, y) ] / h. ∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h.
∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h.
Power Rule
If u = [f(x,y)]n then,
the partial derivative of u with respect to x and y defined as;
ux = n|f(x,y)|n-1∂f/∂x
And uy = n|f(x,y)|n-1∂f/∂y
Lim as x approaches-2
3/2
Lim as x approaches-2
Infinity
Lim as x approaches-1+
0
Lim as x approaches-1-
1
Lim as x approaches-1
0
Lim as x approaches infinity
1
0/0
Cancel common factors
/0
Infinity if +/+ or -/-
