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The study guides havent been updated to the recent changes on the PCAT test (since October 2004). The publishers dont even provide us a supplement! I have been a chemical engineer for five years and now applying for pharmacy schools. Because I took both Oct. and Nov. PCAT tests recently, a friend of mine asked me to advise her on Calculus. I made an outline for her and some of you might be interested it. There are about 10/58 questions related to Calculus. You could use any Calculus textbook to review the items below.
I/ Vectors:
1) Vector a = (a1, a2) is a two-dimensional vector with real numbers a1 and a2.
Features: As a vector is moved parallel to its original one, its coordinates a1 and a2 are unchanged.
2) Given the points A(x1, y1) and B(x2, y2), the vector a representing vector AB is
vector a = (x2-x1, y2-y1)
3) The length of the vector a(a1, a2) is
a = sqrt (sq a1 + sq a2)
The length of vector AB from A(x1, y1) to B(x2, y2) is
AB = sqrt (sq(x2-x1) + sq(y2-y1))
4) Vector addition: vector a = (a1,a2) and vector b = (b1,b2)
vector a + (or -) vector b = (a1 +(-) b1, a2 +(-) b2)
You should know its geometrical representations (Triangle Law and Parallelogram Law)
5) Multiplication of a vector by a scalar (real number)
vector a = (a1, a2) and c is a scalar then c*(vector a) = (c*a1, c*a2)
6) The dot product of vector a = (a1, a2) and vector b = (b1, b2) is a real number
(vector a)*(vector b) = a1*b1 + a2*b2
The product of three vectors is a vector (because the product of first two vectors is a real number or scalar!).
II/Composite function f[g(x)]
ex: Given f(x) = sq(x+4) + 2x + 3 and g(x) = x+1, find f[g(x)]
f[g(x)] = sq[(x+1)+4] + 2(x+1) + 3 {treat g(x) as old x}
= sq(x+5) + 2(x+1) + 3 = (sq x +10*x + 25) + (2x + 2) + 3
= sq x + 12x + 30
III/ Limits:
lim (x to a) of f(x)/g(x)
If both f(a) and g(a) equal zero, then we eliminate the factors (x-a) at both numerator and denominator or use LHospital rule (derivative)
IV/ Derivatives: The first derivative df(x)/dx is
1) The slope of a tangent
2) A rate of change (velocity)
Formulas need to be memorized are:
d(x^n)/dx = n*x^(n-1)
d(u+(or-)v)/dx = du/dx +(-) dv/dx
d(u/v)/dx = (vdu/dx - udv/dx)/v^2
with u = u(x) and v = v(x)
V/ Integral:
S f(x)dx (from a to b) = the area under the graph f(x) from a to b.
Formulas need to be memorized are:
S f(x)dx (from a to b) = F(b) F(a) with F(x) = f(x)
S f(x)dx (from a to b) = - S f(x)dx (from b to a)
S dx = x + c
S x^n dx = (x^(n+1))/(n+1) + c with n # -1
S (1/x)dx = ln(x) + c
S [f(x) + g(x)]dx = S f(x)dx + S g(x)dx
Note: u(x), v(x), f(x), and g(x) are simple functions such as a*x^2 + b*x + c. Dont study trigonometric and root power functions. They are very complicated with derivatives and integrals.
Comments: Why is Calculus needed for studying the pharmacy? You could derive the rate of chemical reactions, find the half life of a molecule, and read the science publications.
If you know the fundamental Calculus above, then you are well prepared for the PCAT test. You would never have enough time to finish the whole quantitative section, so dont freak out this Calculus. Merry Christmas!
I/ Vectors:
1) Vector a = (a1, a2) is a two-dimensional vector with real numbers a1 and a2.
Features: As a vector is moved parallel to its original one, its coordinates a1 and a2 are unchanged.
2) Given the points A(x1, y1) and B(x2, y2), the vector a representing vector AB is
vector a = (x2-x1, y2-y1)
3) The length of the vector a(a1, a2) is
a = sqrt (sq a1 + sq a2)
The length of vector AB from A(x1, y1) to B(x2, y2) is
AB = sqrt (sq(x2-x1) + sq(y2-y1))
4) Vector addition: vector a = (a1,a2) and vector b = (b1,b2)
vector a + (or -) vector b = (a1 +(-) b1, a2 +(-) b2)
You should know its geometrical representations (Triangle Law and Parallelogram Law)
5) Multiplication of a vector by a scalar (real number)
vector a = (a1, a2) and c is a scalar then c*(vector a) = (c*a1, c*a2)
6) The dot product of vector a = (a1, a2) and vector b = (b1, b2) is a real number
(vector a)*(vector b) = a1*b1 + a2*b2
The product of three vectors is a vector (because the product of first two vectors is a real number or scalar!).
II/Composite function f[g(x)]
ex: Given f(x) = sq(x+4) + 2x + 3 and g(x) = x+1, find f[g(x)]
f[g(x)] = sq[(x+1)+4] + 2(x+1) + 3 {treat g(x) as old x}
= sq(x+5) + 2(x+1) + 3 = (sq x +10*x + 25) + (2x + 2) + 3
= sq x + 12x + 30
III/ Limits:
lim (x to a) of f(x)/g(x)
If both f(a) and g(a) equal zero, then we eliminate the factors (x-a) at both numerator and denominator or use LHospital rule (derivative)
IV/ Derivatives: The first derivative df(x)/dx is
1) The slope of a tangent
2) A rate of change (velocity)
Formulas need to be memorized are:
d(x^n)/dx = n*x^(n-1)
d(u+(or-)v)/dx = du/dx +(-) dv/dx
d(u/v)/dx = (vdu/dx - udv/dx)/v^2
with u = u(x) and v = v(x)
V/ Integral:
S f(x)dx (from a to b) = the area under the graph f(x) from a to b.
Formulas need to be memorized are:
S f(x)dx (from a to b) = F(b) F(a) with F(x) = f(x)
S f(x)dx (from a to b) = - S f(x)dx (from b to a)
S dx = x + c
S x^n dx = (x^(n+1))/(n+1) + c with n # -1
S (1/x)dx = ln(x) + c
S [f(x) + g(x)]dx = S f(x)dx + S g(x)dx
Note: u(x), v(x), f(x), and g(x) are simple functions such as a*x^2 + b*x + c. Dont study trigonometric and root power functions. They are very complicated with derivatives and integrals.
Comments: Why is Calculus needed for studying the pharmacy? You could derive the rate of chemical reactions, find the half life of a molecule, and read the science publications.
If you know the fundamental Calculus above, then you are well prepared for the PCAT test. You would never have enough time to finish the whole quantitative section, so dont freak out this Calculus. Merry Christmas!
