Calculus - Grade 12 - Unit 2 (start at lesson 7)

find the derivative of f(x) = 3^(x²-5x)

1. find g ' (x) 2. sub in g ' (x) into the chain rule 3. answer: f ' (x) = 3^(x²-5x)ln3(2x-5)

what is the derivative of f(x) = a^x

f '(x) = a^xlna (a to the power of x, times ln(a)), or f '(x) = kf(x) (where k = lna)

determine the slope of the tangent to the curve f(x) = 5^x at the point (1, 5)

f '(x) = a^xlna = 5^xln(5), then input x = 1 and y = 5, making the equation 5ln(5) ≈ 8.04 (note that x is subbed in for f '(x), and y is subbed in for a^x = f(x))

what is the derivative of f(x)

f '(x) = ka^x

what is the resulting number equal to that is placed after the squiggly ≈ sign?

it is the slope of the tangent

what is unique about every point on the graph of y = e^x

the value of the y-coordinate is the same as the slope of the tangent at that point

how can you rewrite y = 2x ÷ ex²

y = 2x (1 ÷ ex²), or y = 2xe^(-x²)

what happens on the computer graph when m is less than one versus greater than one?

when m is less than one the line makes a curved L shape approaching zero, when m is greater than one the line is mirrored on the other side of the y-axis (making a backwards L where it approaches zero the more negative it gets)

how would you graph f(x) = ln(2x)

find y = f(x) for the table using the calculator (eg. ln 2(0.1) = ln(0.2) = -1.61)

how can you confirm the value of k?

graph the function and find the slope of the tangent to the curve at point (0, 1) for any graph

what special property does the natural exponential function have?

it is equal to its own derivative (or the slope of the tangent given in the graph)

what is the value 'k' equal to for the derivative function f '(x) = kf(x)

k is equal to the slope of the exponential function at the point with x-coordinate = 0 (the point 0, 1)

if you have f '(x) = 3^(x²+1)ln3(2x), how would you arrange it?

move the multiplication to the front, eg. (2x)(3^(x²+1)ln3

how can you rewrite y = 2x ÷ e^x²

rewrite it as y = 2xe^(-x²)

what is the rule for finding the derivative of equations that have 'e' as a base? (eg. e^(-x²))

the derivative of the whole equation = the original function × the derivative of the exponent (eg. e^(-x²) × (-2x) = -2xe^-x²)

if you are adding 6x + (-2xe^(x²), where does the e go?

the e is moved out front of whatever equation it is attached to eg. 6x - e^(x²) × 2x

the graph of f(x) = a^x goes through what point?

the graph goes through point (0,1) since a^0 = 1

what happens to f(x) and f '(x) when x gets bigger?

the value of f(x) gets bigger and the derivative also gets bigger

when do you use the wiggly equal sign?

when finding the derivative of an equation (eg. 3^x) using a^xlna

how would you complete the following table: Function = f(x) = 1.7^x Table: Point (x, y) Value of X (given) Value of f(x) = 1.7^x k = f '(x) ÷ f(x)

(move the m value on the graph to equal 1.7) 1. find the point on the graph using the given value of x 2. input y-value for f(x) 3. estimate the derivative f '(x) using the slope of the tangent provided on the graph 4. then find the mean of k to estimate the value of k

how would you determine the instantaneous rate of change of the function f(x) = 3^x at the point (2, 9)?

1. find f '(x) using a^xlna 2. sub in the point (2, 9) where f '(x) becomes f '(2) and 3^x becomes 9 (because f(x) = y) 3. solve the equation using the calculator 4. add the wiggly equal sign between 9ln3 ≈ 9.8875 (be careful to press the number, then ln, and not the other way around)

how do you find the equation of the tangent to a function? (eg. 3^(x²+1)-x² at point 1, 8)

1. find the derivative of the functions 2. for f(x) = 3^(x²+1)-x², find the derivative of the exponent (g(x)) = 2x 3. sub in the values found at g '(x) into the chain rule 4. use the difference rule because the two original functions were minus each other 5. sub in the x-value of point (1, 8) to the resulting difference equation and solve 6. write the resulting number using the squiggly equal sign ≈ 17.77 7. put the info into y = mx + b form 8. now sub in the x and y values given to solve for b 9. finally, write the whole equation in y = mx + b form using 17.77 as the slope

how would you find the derivative of y = e^(3x)+2

1. separate f(x) and g(x) 2. find f '(x) and g '(x) 3. f '(x) = 3e^(3x), g '(x) = 0 4. for a basic polynomial, simply add the resulting terms (3e^(3x) + 0) 5. isolate the e-value with the power = e^(3x) × 3

what are base-10 logarithms?

Eg. 10² = 100 = log 100, written log₁₀¹⁰⁰ = 2 (what power do you have to raise 10 to to get 100) (x is the base-10 logarithm of y: x = log y)

what does 'e' mean?

Euler's constant ≈ 2.71828, also called the natural exponential function f(x) = e^x

what is the rule for exponential functions of the form f(x) = a^x

If f(x) = a^x and is an exponential function where a is greater than 1, the derivative is f '(x) = kf(x)

exponential function

any function of the form f(x) = a^x which is typically increasing and illustrates exponential growth

if you have 18ln(3)-2, which part do you solve first?

do the 3 × ln first and THEN multiply by 18 and subtract 2

how would you indicate the natural logarithm of y = e^x at the point (-0.69, 0.5)?

e^-0.69 = 0.5, and ln(0.5) = -0.69

what is the derivative of e^x

e^x is the derivative

how do you estimate the limit for a value of a in an exponential function?

estimate the limit by lim(h→0) [a^h-1] ÷ h, making each value of h closer and closer to 1 until you find a likely limit (or slope of the tangent) which would also be equal to f '(x) for the function

which type of function is the only type for which the derivative is proportional to the original function?

exponential functions

chain rule for exponential functions

f '(x) = a^(g(x))lnag'(x) (when f(x) = a^(g(x))